As an illustration of the mediation process that occurs all around us, experiments with quantum systems, referred to as "The Great Smokey Dragon" by John Wheeler, are depicted in Figure 5. Here, a probing particle is injected by first undergoing a classical increase in its energy. It's phase offset, $\phi_p \equiv \phi_o$ evolves with the wavevector as the particle behaves classically to an accelerating force. Upon interacting with a quantum system component with phase $\phi_q$, phase mediation progresses until the two components reach an equilibrium phase. Then another probe, this time radiation, is used to extract the particle from the system and detect it. As depicted, the particle ejector and detector are at equilibrium with the environment, which by convention has a phase offset of $\phi_c = 0^{\circ}$.

Based on this characterization of phase mediation, it is natural to ask what would happen if there is a short-range force field characterizing the surface of a particle. Would that impact how they collide?

This would depend on the details of $\Gamma$ and $\phi'_F$. To couple in the strength of a force, it is postulated that $\Gamma$ takes the following form:

$$ \frac{\pi}{2}\Gamma = \tan^{-1}\left(\frac{\lambda + \lambda_o}{\lambda_F}\right) \quad (143) $$

Just as with the photoelectric effect, the dependency is not on the amplitude of the force, but rather, the base frequency or wavelength associated with the interaction. This has the proper range and limiting behavior. With $\frac{\lambda}{\lambda_F} \ll 1$, the energy of the particle is too high for mediation to be effective since $\epsilon \to \epsilon_o$:

$$ \epsilon_o = \frac{2}{\pi} \tan^{-1} \left( \frac{\lambda_o}{\lambda_F} \right) \cos \bar{\phi}_{c1} \quad (144) $$

In this limit, there is no effective damping of the phase offset due to chaos. When $\frac{\lambda}{\lambda_F}$ or $\frac{\lambda_o}{\lambda_F} \to \infty$, the approximation in (136) is exact and the coupling, $\epsilon = \cos \bar{\phi}_{c1} = 1$, ensures no further mediation takes place.

Given the energy associated with accelerators, a particle's wavelength is likely to be in a range where $\epsilon \to \epsilon_o$ dominates until the approach is subatomic. As such, $\phi'_F$ will be mediated by a long-range force first before interacting with the short-range force. Given $\lambda_F \to 0$ for the short-range force as a particle approaches the other, $\epsilon \to 1$ after the long-range mediation has taken place.

Thus, there is a point at which $\phi'_F$'s value is already mediated and neither influenced by the long-range or short-range forces attributed to particle-particle interactions. From (131), its state takes the form $\langle\phi'_F|x\rangle = \sin\phi$ according to the mediation analysis in this section and others. However, (142) specifies that $\phi'_F$ must go to zero at the origin of the particle. Given significant mediation is not possible at close range based on the arguments just made, it must have happened that the long-range mediation chose a phase offset within $\phi$ such that $\langle 0|\phi \rangle = 0$. In other words, $\phi'_F = \phi = 0$ at the origin of both particles, hence $\sin\phi = \cos\phi_e = 0$.

<figure><img src="image_8.png" /><figcaption>FIG. 6. Standard wave depiction of Young's double slit experiment using Huygen's principle. This requires the concept that wavetrains of indefinite length permit a waveform to interfere with itself along different paths, involving different stretches of time. Experimentally, interference patterns indicate that particles are still detected at the ideal minima of the lowest order interference nodes, referred to here as the halo effect. The explanation for this suggests the varying width of the carrier is in part responsible as well as the modified Huygen's principle where gratings with more than two slits introduce slight phase differences of the particle's transverse wave, which is the most noticeable about the center slit.</figcaption></figure>

Physically, this might suggest that the fields of indistinguishable particles overlapping one another aren't capable of distinguishing between source and target. Thus, neither exists and the force associated with the phase projection (115) is zero while the transverse extension of the phase bubble, dependent upon $\cos\phi$, expands to its maximum transverse size. Hence, no collision occurs. This conclusion does not apply to composite particles because the reach of short-range forces away from the particle's center may be significant.

## VII. YOUNG'S DOUBLE SLIT EXPERIMENT

The quintessential example of quantum behavior and particle-wave duality is Young's double-slit experiment. The standard depiction of interference is shown in Figure 6. Here, according to Huygen's principle, wavelets emitted at different points in time converge to define the phase at any given point. This creates the impression that waves permeate all of space according to the $\frac{1}{\sqrt{L}}$ normalization of the amplitudes. However, in order for this concept to work, wavetrains have to extend for an indefinite length. In two-slit interference, for example, the leading edge of a wavetrain at one slit interacts with its trailing edge at the other slit. Though robust for wave characterizations, its resolution with particle behavior has only been characterized by probabilities.

This theory depicts the process in a different way, starting with the standard representation of $\phi$ using two circular wavelike terms emanating from two very narrow
As an illustration of the mediation process that occurs all around us, experiments with quantum systems, referred to as "The Great Smokey Dragon" by John Wheeler, are depicted in Figure 5. Here, a probing particle is injected by first undergoing a classical increase in its energy. It's phase offset, $\phi_p \equiv \phi_o$ evolves with the wavevector as the particle behaves classically to an accelerating force. Upon interacting with a quantum system component with phase $\phi_q$, phase mediation progresses until the two components reach an equilibrium phase. Then another probe, this time radiation, is used to extract the particle from the system and detect it. As depicted, the particle ejector and detector are at equilibrium with the environment, which by convention has a phase offset of $\phi_c = 0^{\circ}$.

Based on this characterization of phase mediation, it is natural to ask what would happen if there is a short-range force field characterizing the surface of a particle. Would that impact how they collide?

This would depend on the details of $\Gamma$ and $\phi'_F$. To couple in the strength of a force, it is postulated that $\Gamma$ takes the following form:

$$ \frac{\pi}{2}\Gamma = \tan^{-1}\left(\frac{\lambda + \lambda_o}{\lambda_F}\right) \quad (143) $$

Just as with the photoelectric effect, the dependency is not on the amplitude of the force, but rather, the base frequency or wavelength associated with the interaction. This has the proper range and limiting behavior. With $\frac{\lambda}{\lambda_F} \ll 1$, the energy of the particle is too high for mediation to be effective since $\epsilon \to \epsilon_o$:

$$ \epsilon_o = \frac{2}{\pi} \tan^{-1} \left( \frac{\lambda_o}{\lambda_F} \right) \cos \bar{\phi}_{c1} \quad (144) $$

In this limit, there is no effective damping of the phase offset due to chaos. When $\frac{\lambda}{\lambda_F}$ or $\frac{\lambda_o}{\lambda_F} \to \infty$, the approximation in (136) is exact and the coupling, $\epsilon = \cos\bar{\phi}_{c1} = 1$, ensures no further mediation takes place.

Given the energy associated with accelerators, a particle's wavelength is likely to be in a range where $\epsilon \to \epsilon_o$ dominates until the approach is subatomic. As such, $\phi'_F$ will be mediated by a long-range force first before interacting with the short-range force. Given $\lambda_F \to 0$ for the short-range force as a particle approaches the other, $\epsilon \to 1$ after the long-range mediation has taken place.

Thus, there is a point at which $\phi'_F$'s value is already mediated and neither influenced by the long-range or short-range forces attributed to particle-particle interactions. From (131), its state takes the form $\langle\phi'_F|x\rangle = \sin\phi$ according to the mediation analysis in this section and others. However, (142) specifies that $\phi'_F$ must go to zero at the origin of the particle. Given significant mediation is not possible at close range based on the arguments just made, it must have happened that the long-range mediation chose a phase offset within $\phi$ such that $\langle 0|\phi \rangle = 0$. In other words, $\phi'_F = \phi = 0$ at the origin of both particles, hence $\sin\phi = \cos\phi_e = 0$.

<figure><img src="image_7.png" /><figcaption>FIG. 6. Standard wave depiction of Young's double slit experiment using Huygen's principle. This requires the concept that wavetrains of indefinite length permit a waveform to interfere with itself along different paths, involving different stretches of time. Experimentally, interference patterns indicate that particles are still detected at the ideal minima of the lowest order interference nodes, referred to here as the halo effect. The explanation for this suggests the varying width of the carrier is in part responsible as well as the modified Huygen's principle where gratings with more than two slits introduce slight phase differences of the particle's transverse wave, which is the most noticeable about the center slit.</figcaption></figure>

Physically, this might suggest that the fields of indistinguishable particles overlapping one another aren't capable of distinguishing between source and target. Thus, neither exists and the force associated with the phase projection (115) is zero while the transverse extension of the phase bubble, dependent upon $\cos\phi$, expands to its maximum transverse size. Hence, no collision occurs. This conclusion does not apply to composite particles because the reach of short-range forces away from the particle's center may be significant.

## VII. YOUNG'S DOUBLE SLIT EXPERIMENT

The quintessential example of quantum behavior and particle-wave duality is Young's double-slit experiment. The standard depiction of interference is shown in Figure 6. Here, according to Huygen's principle, wavelets emitted at different points in time converge to define the phase at any given point. This creates the impression that waves permeate all of space according to the $\frac{1}{\sqrt{L}}$ normalization of the amplitudes. However, in order for this concept to work, wavetrains have to extend for an indefinite length. In two-slit interference, for example, the leading edge of a wavetrain at one slit interacts with its trailing edge at the other slit. Though robust for wave characterizations, its resolution with particle behavior has only been characterized by probabilities.

This theory depicts the process in a different way, starting with the standard representation of $\phi$ using two circular wavelike terms emanating from two very narrow
slits:

$$ \langle \vec{x} | \Sigma_i \mathbf{A}_i \cdot \phi_i \rangle = \mathbf{A}_\phi \cdot \left( \frac{e^{+i\hat{\chi}_\pm(\vec{k}_{1c} \cdot \vec{x}_1 - \phi_o)}}{\sqrt{|\vec{x}_1|}} + \frac{e^{+i(\vec{k}_{2c} \cdot \vec{x}_2 - \phi_o)}}{\sqrt{|\vec{x}_2|}} \right) \quad (145) $$

Here, the squares of the amplitudes, characterizing probability, are normalized on the perimeter of a circle, $2\pi|\vec{x}_i|$, with $\sqrt{2\pi}$ incorporated into $\mathbf{A}_\phi$. It is assumed that the quanta travels through the left slit, hence positional unit vector $\hat{x}_1$ is generated by the $\hat{\chi}_\pm$ operator, which is not present for the second term because this does not characterize the quanta's position.

Up until now, single particle solutions have been examined, represented by real or imaginary $\mathbf{A}_\phi$ coefficients. In this analysis, there is an unlimited number of particles with arbitrary phase offsets, hence $\mathbf{A}_\phi = A \pm iA$. Using (16), this yields a probability in the following form:

$$ P(\vec{x}_1, \vec{x}_2) = NA^2 \left( \frac{1}{|\vec{x}_1|} \cos \left( \frac{\pi}{2} \hat{x}_1 \cdot \hat{x}_o - \vec{k}_{1c} \cdot \vec{x}_1 + \phi_o \right) \right) \\ + \frac{1}{|\vec{x}_2|} \cos \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right)^2 \\ + NA^2 \left( \frac{1}{|\vec{x}_1|} \sin \left( \frac{\pi}{2} \hat{x}_1 \cdot \hat{x}_o - \vec{k}_{1c} \cdot \vec{x}_1 + \phi_o \right) \right) \\ + \frac{1}{|\vec{x}_2|} \sin \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right)^2 \quad (146) $$

where $\cos(\alpha) = \cos(-\alpha)$ is used to rearrange terms introduced by (2). Assuming $\hat{x} \parallel \hat{x}_o$ after enough distance is traveled away from the slits, a condition examined in more detail further on, the standard approximation, $|\vec{x}_1| = |\vec{x}_2| \approx L$, is applied:

$$ P(\vec{x}_1, \vec{x}_2) \approx \frac{NA^2}{L} \left( \cos \left( \vec{k}_{1c} \cdot \vec{x}_1 - \phi_o \right) + \cos \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right) \right)^2 + \frac{NA^2}{L} \left( \sin \left( \vec{k}_{1c} \cdot \vec{x}_1 - \phi_o \right) + \sin \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right) \right)^2 \quad (147) $$

Here, $\phi_o$ is associated with the carrier envelope, determined by boundary conditions, which plays no role in the conventional analysis presented below, but it will be shown that it is important for characterizing apparent decoherence. The final step for this standard approach to the problem is to apply the following trigonometric identities to each of the squared terms:

$$ \begin{aligned} \cos\alpha + \cos\beta &= 2\cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \\ \sin\alpha + \sin\beta &= 2\sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \quad (148) \end{aligned} $$

Using the Pythagorean identity, the combination of replacements reduces to the following well-known expres-

sion at large distances:

$$ \begin{aligned} P(\vec{x}_1, \vec{x}_2) &\approx \frac{4A^2}{L} \cos^2 \left( \frac{\vec{k}_{1c} \cdot \vec{x}_1 - \vec{k}_{2c} \cdot \vec{x}_2}{2} \right) \\ &= \frac{4A^2}{L} \cos^2 \left( \frac{k_e(x_1 - x_2)}{2} \right) \quad (149) \end{aligned} $$

Since the dynamics of the particle outside the grating satisfies the free particle equation (40), the same approach for determining $\vec{k}$ in $\mathbb{R}$ can be used by taking the derivative of $P$, however given the relationship between $\phi$ and $\nu$ brought out by the study of superposition in Section III, the following identification of the frequency is readily made:

$$ \nu(\vec{x}_1, \vec{x}_2) = \nu_a \cos^2 \left( \frac{k_e(x_1 - x_2)}{2} \right) \quad (150) $$

Then (40) can be solved assuming an exchange energy in the form of (42). To do this, the position vector $\vec{x}_2$ is expressed in terms of the position vector of the quanta, $\vec{x}_1$, using the separation of slits represented by vector $\vec{a}$:

$$ \vec{x}_2 = \vec{x}_1 - \vec{a} \quad (151) $$

The expression for $\nu(\vec{x})$ then becomes:

$$ \begin{aligned} \nu(\vec{x}_1) &= \nu_a \cos^2 (\phi_{12}(\vec{x}_1)) \\ \phi_{12}(\vec{x}_1) &= \frac{1}{2} \left( (\vec{k}_{1c} - \vec{k}_{2c}) \cdot \vec{x}_1 + \vec{k}_{2c} \cdot \vec{a} \right) \quad (153) \end{aligned} $$

with the solution to (40) given by:

$$ \begin{aligned} \lambda(\vec{x}_1) &= \lambda_a \tan (\phi_{12}(\vec{x}_1)) \\ \alpha_{\parallel}^{-1} \nu_a &= \frac{(\vec{k}_{1c} - \vec{k}_{2c}) \cdot \hat{x}_1}{2k_a} \nu_a \end{aligned} \quad (154) $$

where it is assumed that $\hat{x}_o \parallel \hat{x}_1$.

The analysis presented so far is depicted in Figure 7. Just as for Huygen's original principle, wavelets from different points of the wavefront reach the quanta's location after the same amount of time has elapsed. However, as a modification to Huygen's principle, wavefronts are phase-balanced and formed by the carriers of a statistical ensemble of particles, e.g. treated as planar or spherical in shape due to their compression in comparison to an envelope. This results in the generation of wavelets with different velocities. It is how the phase bubble concept replaces the notion of wavetrains and can cast wavelets as imaginary, since the propagation of their superposition actually occurs within the carrier as it collapses, expands, and travels forward, according to the properties of a modulated transverse wave.

To demonstrate that the imaginary wavelets arrive at the quanta at the same time without the use of wave-trains, the velocity ratio $v_1/c$ assigned to the quanta's path
slits:

$$ \langle \vec{x} | \Sigma_i \mathbf{A}_i \cdot \phi_i \rangle = \mathbf{A}_\phi \cdot \left( \frac{e^{+i\hat{\chi}_\pm(\vec{k}_{1c} \cdot \vec{x}_1 - \phi_o)}}{\sqrt{|\vec{x}_1|}} + \frac{e^{+i(\vec{k}_{2c} \cdot \vec{x}_2 - \phi_o)}}{\sqrt{|\vec{x}_2|}} \right) \quad (145) $$

Here, the squares of the amplitudes, characterizing probability, are normalized on the perimeter of a circle, $2\pi|\vec{x}_i|$, with $\sqrt{2\pi}$ incorporated into $\mathbf{A}_\phi$. It is assumed that the quanta travels through the left slit, hence positional unit vector $\hat{x}_1$ is generated by the $\hat{\chi}_\pm$ operator, which is not present for the second term because this does not characterize the quanta's position.

Up until now, single particle solutions have been examined, represented by real or imaginary $\mathbf{A}_\phi$ coefficients. In this analysis, there is an unlimited number of particles with arbitrary phase offsets, hence $\mathbf{A}_\phi = A \pm iA$. Using (16), this yields a probability in the following form:

$$ P(\vec{x}_1, \vec{x}_2) = NA^2 \left( \frac{1}{|\vec{x}_1|} \cos \left( \frac{\pi}{2} \hat{x}_1 \cdot \hat{x}_o - \vec{k}_{1c} \cdot \vec{x}_1 + \phi_o \right) \right) \\ + \frac{1}{|\vec{x}_2|} \cos \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right)^2 \\ + NA^2 \left( \frac{1}{|\vec{x}_1|} \sin \left( \frac{\pi}{2} \hat{x}_1 \cdot \hat{x}_o - \vec{k}_{1c} \cdot \vec{x}_1 + \phi_o \right) \right) \\ + \frac{1}{|\vec{x}_2|} \sin \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right)^2 \quad (146) $$

where $\cos(\alpha) = \cos(-\alpha)$ is used to rearrange terms introduced by (2). Assuming $\hat{x} \parallel \hat{x}_o$ after enough distance is traveled away from the slits, a condition examined in more detail further on, the standard approximation, $|\vec{x}_1| = |\vec{x}_2| \approx L$, is applied:

$$ P(\vec{x}_1, \vec{x}_2) \approx \frac{NA^2}{L} \left( \cos \left( \vec{k}_{1c} \cdot \vec{x}_1 - \phi_o \right) + \cos \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right) \right)^2 + \frac{NA^2}{L} \left( \sin \left( \vec{k}_{1c} \cdot \vec{x}_1 - \phi_o \right) + \sin \left( \vec{k}_{2c} \cdot \vec{x}_2 - \phi_o \right) \right)^2 \quad (147) $$

Here, $\phi_o$ is associated with the carrier envelope, determined by boundary conditions, which plays no role in the conventional analysis presented below, but it will be shown that it is important for characterizing apparent decoherence. The final step for this standard approach to the problem is to apply the following trigonometric identities to each of the squared terms:

$$ \begin{aligned} \cos\alpha + \cos\beta &= 2\cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \\ \sin\alpha + \sin\beta &= 2\sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \quad (148) \end{aligned} $$

Using the Pythagorean identity, the combination of replacements reduces to the following well-known expres-

sion at large distances:

$$ \begin{aligned} P(\vec{x}_1, \vec{x}_2) &\approx \frac{4A^2}{L} \cos^2 \left( \frac{\vec{k}_{1c} \cdot \vec{x}_1 - \vec{k}_{2c} \cdot \vec{x}_2}{2} \right) \\ &= \frac{4A^2}{L} \cos^2 \left( \frac{k_e(x_1 - x_2)}{2} \right) \quad (149) \end{aligned} $$

Since the dynamics of the particle outside the grating satisfies the free particle equation (40), the same approach for determining $\vec{k}$ in $\mathbb{R}$ can be used by taking the derivative of $P$, however given the relationship between $\phi$ and $\nu$ brought out by the study of superposition in Section III, the following identification of the frequency is readily made:

$$ \nu(\vec{x}_1, \vec{x}_2) = \nu_a \cos^2 \left( \frac{k_e(x_1 - x_2)}{2} \right) \quad (150) $$

Then (40) can be solved assuming an exchange energy in the form of (42). To do this, the position vector $\vec{x}_2$ is expressed in terms of the position vector of the quanta, $\vec{x}_1$, using the separation of slits represented by vector $\vec{a}$:

$$ \vec{x}_2 = \vec{x}_1 - \vec{a} \quad (151) $$

The expression for $\nu(\vec{x})$ then becomes:

$$ \begin{aligned} \nu(\vec{x}_1) &= \nu_a \cos^2(\phi_{12}(\vec{x}_1)) \\ \phi_{12}(\vec{x}_1) &= \frac{1}{2} \left( (\vec{k}_{1c} - \vec{k}_{2c}) \cdot \vec{x}_1 + \vec{k}_{2c} \cdot \vec{a} \right) \quad (153) \end{aligned} $$

with the solution to (40) given by:

$$ \begin{aligned} \lambda(\vec{x}_1) &= \lambda_a \tan(\phi_{12}(\vec{x}_1)) \\ \alpha_{\parallel}^{-1} \nu_a &= \frac{(\vec{k}_{1c} - \vec{k}_{2c}) \cdot \hat{x}_1}{2k_a} \nu_a \end{aligned} \quad (154) $$

where it is assumed that $\hat{x}_o \parallel \hat{x}_1$.

The analysis presented so far is depicted in Figure 7. Just as for Huygen's original principle, wavelets from different points of the wavefront reach the quanta's location after the same amount of time has elapsed. However, as a modification to Huygen's principle, wavefronts are phase-balanced and formed by the carriers of a statistical ensemble of particles, e.g. treated as planar or spherical in shape due to their compression in comparison to an envelope. This results in the generation of wavelets with different velocities. It is how the phase bubble concept replaces the notion of wavetrains and can cast wavelets as imaginary, since the propagation of their superposition actually occurs within the carrier as it collapses, expands, and travels forward, according to the properties of a modulated transverse wave.

To demonstrate that the imaginary wavelets arrive at the quanta at the same time without the use of wave-trains, the velocity ratio $v_1/c$ assigned to the quanta's path
<figure><img src="image_9.png" /><figcaption>FIG. 7. Modified Huygen's principle for a plane wave. Here, imaginary wavelets emanating from a phase-balanced wavefront contribute only to the phase at the location of the quanta. No wavetrain is present. Two contributions that cancel each other are shown for $k_{2c}$ and $k'_{2c}$, which point in opposite directions since the sign of the phase is attached to them. In reality, the phase is determined by the propagation dynamics of a modulated transverse wave.</figcaption></figure>

and $\frac{v_2}{c}$ assigned to a separate wavevector on the wavefront are given by the following:

$$ \frac{v_2}{c} = k_c \cos \theta_v \qquad (155) $$

$$ \frac{v_1}{c} = k_c \cos \left( \theta_v + \cos^{-1} \left( \frac{x_1}{x_2} \right) \right) + k_c \frac{a}{x_2} \sin \theta_v \qquad (156) $$

where $v_2 > v_1$ due to $\theta_v \equiv \cos^{-1}(\frac{v_2}{c})$ and $\vec{k}_{2c}$ vector's closer alignment with $\vec{x}_1$ compared to $\vec{k}_{1c}$ vector's alignment. This is consistent with their direction relative to the average velocity of propagation as depicted in Figure 1. The ratio of distances for the respective paths define the angle between the two vectors $\vec{k}_{1c}$ and $\vec{k}_{2c}$, with the second term in (156) necessary to compensate, as seen in the limit when $v_1 = 0$, $x_1 = 0$, and $x_2 = a$.

Expanding out the term for $v_1$,

$$ \frac{v_1}{c} = k_c \frac{x_1}{x_2} \cos \theta_v - k_c \sin \left( \cos^{-1} \left( \frac{x_1}{x_2} \right) \right) \sin \theta_v + k_c \frac{a}{x_2} \sin \theta_v = k_c \frac{x_1}{x_2} \cos \theta_v \qquad (157) $$

Its reduction is possible due to:

$$ \frac{a}{x_2} = \sin \left( \cos^{-1} \left( \frac{x_1}{x_2} \right) \right) \qquad (158) $$

Hence, using (155) and (157), the time required for the two wavelets to travel their respective distances satisfies:

$$ t = \frac{v_1}{x_1} = \frac{v_2}{x_2} \qquad (159) $$

The contribution to the phase at the quanta is equivalent to the method of Huygen's original principle with one exception. As there are no wavetrains, the phases contributed by different points on the wavefront do not all superimpose constructively for the case of a freely propagating particle. Instead, half of the wavefront contributes negative phases, and the other half positive, separated by the trajectory of the quanta due to the wavefront's manifestation as the varying amplitude of the carrier. Just as argued in Feynman's path integral approach, the summation of phases over all paths cancel for a freely propagating particle except for the one assigned to the classical, i.e. quanta's, trajectory.

However, if part of the wavefront is blocked, say by two slits, then a net contribution to phase occurs from a component of the wavefront that is not on the quanta's path. Applying this modified principle to a grating with more than two slits produces an interesting result. Superposition of all possible trajectories of incident particles somewhat dampen the effects of negative and positive phases at the slits on either side of the one the quanta passes through, except for the center slit. Effectively, the positive and negative phase shifts about the center slit can have a slight impact near the center region of a screen, depending on the transverse amplitude, with partially washed-out interference, referred to as the halo effect in Figure 6. Another cause is due to the varying amplitude of the transverse wave according to plots in Figure 4. Since there is no transverse amplitude observed at certain points in space, a certain percentage of particles with a phase offset $\phi_o$ different from others will pass through a single slit without interacting with another slit. Only diffraction is at play for such particles.

Examining the two slit case further, $\hat{x} \cdot \hat{x}_o = 1$ continues to hold true as the quanta moves inside the grating initially, but as the carrier envelope gets pinned by boundary conditions, the quanta is free to acquire a three-dimensional velocity due to interactions with the sides of the grating slits. As shown in Figure 8, the grating can be thought of as a short waveguide and the carrier wave propagates through with boundary conditions flattening the wave amplitude as an extended node inside the regions that have mass. Depicted is the appearance of a brief tunneling exchange between slits, which has a low probability. The resulting trajectory of the quanta upon reaching the other side of the grating depends on random variables leading to diffraction from the finite width of the grating, which is ignored in this analysis. However, upon exit the effect of the slit boundaries on orientation become less pronounced as the carrier rotates to be in alignment with the motion of the quanta.

What is only suggested in Figure 8 is how the envelope distorts as the particle travels away from the grating. Though wavevector $k_e$ is a constant, the impact of $x_1 - x_2$ in (149) changes with distance, increasing the effective wavelength $\lambda_e(x_1)$ and decreasing the frequency $\nu_e(x_1)$ such that their product still gives the same velocity, but with increasing separation between probability
<figure><img src="image_8.png" /><figcaption>FIG. 7. Modified Huygen's principle for a plane wave. Here, imaginary wavelets emanating from a phase-balanced wavefront contribute only to the phase at the location of the quanta. No wavetrain is present. Two contributions that cancel each other are shown for $k_{2c}$ and $k'_{2c}$, which point in opposite directions since the sign of the phase is attached to them. In reality, the phase is determined by the propagation dynamics of a modulated transverse wave.</figcaption></figure>

and $\frac{v_2}{c}$ assigned to a separate wavevector on the wavefront are given by the following:

$$ \frac{v_2}{c} = k_c \cos \theta_v \quad (155) $$

$$ \frac{v_1}{c} = k_c \cos \left( \theta_v + \cos^{-1} \left( \frac{x_1}{x_2} \right) \right) + k_c \frac{a}{x_2} \sin \theta_v \quad (156) $$

where $v_2 > v_1$ due to $\theta_v \equiv \cos^{-1}(\frac{v_2}{c})$ and $\vec{k}_{2c}$ vector's closer alignment with $\vec{x}_1$ compared to $\vec{k}_{1c}$ vector's alignment. This is consistent with their direction relative to the average velocity of propagation as depicted in Figure 1. The ratio of distances for the respective paths define the angle between the two vectors $\vec{k}_{1c}$ and $\vec{k}_{2c}$, with the second term in (156) necessary to compensate, as seen in the limit when $v_1 = 0$, $x_1 = 0$, and $x_2 = a$.

Expanding out the term for $v_1$,

$$ \frac{v_1}{c} = k_c \frac{x_1}{x_2} \cos \theta_v - k_c \sin \left( \cos^{-1} \left( \frac{x_1}{x_2} \right) \right) \sin \theta_v \\ + k_c \frac{a}{x_2} \sin \theta_v = k_c \frac{x_1}{x_2} \cos \theta_v \quad (157) $$

Its reduction is possible due to:

$$ \frac{a}{x_2} = \sin \left( \cos^{-1} \left( \frac{x_1}{x_2} \right) \right) \quad (158) $$

Hence, using (155) and (157), the time required for the two wavelets to travel their respective distances satisfies:

$$ t = \frac{v_1}{x_1} = \frac{v_2}{x_2} \quad (159) $$

The contribution to the phase at the quanta is equivalent to the method of Huygen's original principle with one exception. As there are no wavetrains, the phases contributed by different points on the wavefront do not all superimpose constructively for the case of a freely propagating particle. Instead, half of the wavefront contributes negative phases, and the other half positive, separated by the trajectory of the quanta due to the wavefront's manifestation as the varying amplitude of the carrier. Just as argued in Feynman's path integral approach, the summation of phases over all paths cancel for a freely propagating particle except for the one assigned to the classical, i.e. quanta's, trajectory.

However, if part of the wavefront is blocked, say by two slits, then a net contribution to phase occurs from a component of the wavefront that is not on the quanta's path. Applying this modified principle to a grating with more than two slits produces an interesting result. Superposition of all possible trajectories of incident particles somewhat dampen the effects of negative and positive phases at the slits on either side of the one the quanta passes through, except for the center slit. Effectively, the positive and negative phase shifts about the center slit can have a slight impact near the center region of a screen, depending on the transverse amplitude, with partially washed-out interference, referred to as the halo effect in Figure 6. Another cause is due to the varying amplitude of the transverse wave according to plots in Figure 4. Since there is no transverse amplitude observed at certain points in space, a certain percentage of particles with a phase offset $\phi_o$ different from others will pass through a single slit without interacting with another slit. Only diffraction is at play for such particles.

Examining the two slit case further, $\hat{x} \cdot \hat{x}_o = 1$ continues to hold true as the quanta moves inside the grating initially, but as the carrier envelope gets pinned by boundary conditions, the quanta is free to acquire a three-dimensional velocity due to interactions with the sides of the grating slits. As shown in Figure 8, the grating can be thought of as a short waveguide and the carrier wave propagates through with boundary conditions flattening the wave amplitude as an extended node inside the regions that have mass. Depicted is the appearance of a brief tunneling exchange between slits, which has a low probability. The resulting trajectory of the quanta upon reaching the other side of the grating depends on random variables leading to diffraction from the finite width of the grating, which is ignored in this analysis. However, upon exit the effect of the slit boundaries on orientation become less pronounced as the carrier rotates to be in alignment with the motion of the quanta.

What is only suggested in Figure 8 is how the envelope distorts as the particle travels away from the grating. Though wavevector $k_e$ is a constant, the impact of $x_1 - x_2$ in (149) changes with distance, increasing the effective wavelength $\lambda_e(x_1)$ and decreasing the frequency $\nu_e(x_1)$ such that their product still gives the same velocity, but with increasing separation between probability
peaks. The interference stretches out the carrier's amplitude variation with increasing distance, leading eventually to a solution that looks like a plane wave with $P = 1$. Thus, at a far enough distance, the arbitrary phase offsets assigned to the particles start to have a noticeable effect on the measurements.

This behavior seems very much like the broadening of a Schrödinger wavepacket with distance and time. A reason for it is that although the spacetime average of the velocity of the particle remains the same as without interference, the space and time scale over which this average must be taken increases with distance: a fact that lends support to standard quantum theory's treatment of uncertainty in its analysis but as demonstrated further along in this work, accuracy may be impacted for studies where uncertainty is bounded.

To calculate how the carrier direction and quanta realign, it is noted that upon leaving the left grating slit, the quanta has a change in direction while the carrier propagation direction is still perpendicular to the grating. It will be assumed that $\phi$ is dominated by the waveform exiting the left slit due to the $\frac{1}{x}$ normalization, hence to good approximation in that region, (25) applies for the wavelength. Another look at the derivation of (41) using (25) yields the following approximate relationship for $\alpha_\parallel$:

$$ \alpha_{\parallel}^{-1} \approx \alpha_{o}^{-1} - \frac{d}{d\vec{x}_{1}} \left( \frac{\lambda_{a}}{4} \hat{x}_{1} \cdot \hat{x}_{o}(\vec{x}_{1}) \right) \cdot \hat{v} \quad (160) $$

where close to the left slit, $\alpha_o^{-1}$ is given by:

$$ \alpha_o^{-1} \approx \frac{k_e}{k_a} \{x_1 \lesssim a\} \quad (161) $$

while further out from the grating, the asymptotic limit is closer in value, expressing the spreading influence of complete interference with increasing distance:

$$ \alpha_o^{-1} \approx \frac{(\vec{k}_{1c} - \vec{k}_{2c}) \cdot \hat{x}_1}{2k_a} \quad \{x_1 \gtrsim a\} \quad (162) $$

The direction of the carrier's propagation is able to catch up to the direction of motion for the quanta via a restorative influence that simply depends on the derivative of the angle between them. This is expressed more explicitly as follows, noting $\hat{x}_1 = \hat{v}$:

$$ \begin{aligned} \alpha_{\parallel}^{-1} &\approx \alpha_{o}^{-1} - \frac{\lambda_{a}}{4} \frac{d}{d\theta_{o}} \cos(\theta_{1} - \theta_{o}(x_{1})) \frac{d\theta_{o}}{dx_{1}} \\ &= \alpha_{o}^{-1} + \frac{\lambda_{a}}{4} \sin(\theta_{1} - \theta_{o}(x_{1})) \frac{d\theta_{o}}{dx_{1}} \end{aligned} \quad (163) $$

As discussed in the Appendix and mentioned in the introduction to the $\hat{\chi}_\pm$ operator, spin is responsible for keeping $\hat{x}$ and $\hat{x}_o$ aligned so that a particle propagates in the same direction in the absence of an external force. It is thus postulated, in line with spin-orbit coupling, that the phase vector component $k_s$ is determined by the following relationship:

$$ s\hbar = \hbar k_s a \quad (164) $$

<figure><img src="image_10.png" /><figcaption>FIG. 8. Evolving waveform of two-slit interference for transverse wave propagation. Here, the modified Huygen's principle can be used to determine the phase at the evaluation point of the quanta, which obtains the same result as determined from standard theoretical analysis for locations far from the slits. The sketch is only illustrative, stretched out over time, as it does not restrict the waveform to only the phase bubble region in the vicinity of the quanta. Close to the slits, (166) specifies that the carrier rotates about the quanta (center) after it passes through the left slit. Considering two-dimensional analysis of the waveform within the grating, a small tunneling probability is present that would allow the quanta to tunnel between slits, but the occurrence of this is not necessary to characterize two-slit interference.</figcaption></figure>

where the slit separation of $a$ is the moment arm of the effective angular momentum associated with the carrier phase rotation. The phase angle then becomes:

$$ \theta_o(x_1) = \frac{\pi}{2} + k_s x_1 = \frac{\pi}{2} + \frac{s}{a} x_1 \quad (165) $$

This gives the final equation that determines $\alpha_\parallel$:

$$ \alpha_{\parallel}^{-1} \approx \alpha_{o}^{-1} + \frac{s\lambda_{a}}{4a} \sin(\theta_{1} - \theta_{o}(x_{1})) \mathbf{\Theta}(\theta_{1} - \theta_{o}(x_{1})) \quad (166) $$

where $\mathbf{\Theta}$ is the Heaviside function. Thus, the carrier propagation direction is in alignment with the quanta's motion when:

$$ x_1 \gtrsim \frac{a \left| \theta_1 - \frac{\pi}{2} \right|}{s} \quad (167) $$

At this distance, the calculation of the probability in (149) is becoming closer to the correct value.

For the rest of this paper, additional well-known problems will be analyzed with this theory. Explicit consideration of $\hat{\chi}_\pm$ will be dropped in the next few sections to simplify the derivations as phase offsets are determined by boundary conditions.
peaks. The interference stretches out the carrier's amplitude variation with increasing distance, leading eventually to a solution that looks like a plane wave with $P = 1$. Thus, at a far enough distance, the arbitrary phase offsets assigned to the particles start to have a noticeable effect on the measurements.

This behavior seems very much like the broadening of a Schrödinger wavepacket with distance and time. A reason for it is that although the spacetime average of the velocity of the particle remains the same as without interference, the space and time scale over which this average must be taken increases with distance: a fact that lends support to standard quantum theory's treatment of uncertainty in its analysis but as demonstrated further along in this work, accuracy may be impacted for studies where uncertainty is bounded.

To calculate how the carrier direction and quanta realign, it is noted that upon leaving the left grating slit, the quanta has a change in direction while the carrier propagation direction is still perpendicular to the grating. It will be assumed that $\phi$ is dominated by the waveform exiting the left slit due to the $\frac{1}{x}$ normalization, hence to good approximation in that region, (25) applies for the wavelength. Another look at the derivation of (41) using (25) yields the following approximate relationship for $\alpha_\parallel$:

$$ \alpha_\parallel^{-1} \approx \alpha_o^{-1} - \frac{d}{d\vec{x}_1} \left( \frac{\lambda_a}{4} \hat{x}_1 \cdot \hat{x}_o(\vec{x}_1) \right) \cdot \hat{v} \quad (160) $$

where close to the left slit, $\alpha_o^{-1}$ is given by:

$$ \alpha_o^{-1} \approx \frac{k_e}{k_a} \{x_1 \lesssim a\} \quad (161) $$

while further out from the grating, the asymptotic limit is closer in value, expressing the spreading influence of complete interference with increasing distance:

$$ \alpha_o^{-1} \approx \frac{(\vec{k}_{1c} - \vec{k}_{2c}) \cdot \hat{x}_1}{2k_a} \quad \{x_1 \gtrsim a\} \quad (162) $$

The direction of the carrier's propagation is able to catch up to the direction of motion for the quanta via a restorative influence that simply depends on the derivative of the angle between them. This is expressed more explicitly as follows, noting $\hat{x}_1 = \hat{v}$:

$$ \begin{aligned} \alpha_\parallel^{-1} &\approx \alpha_o^{-1} - \frac{\lambda_a}{4} \frac{d}{d\theta_o} \cos(\theta_1 - \theta_o(x_1)) \frac{d\theta_o}{dx_1} \\ &= \alpha_o^{-1} + \frac{\lambda_a}{4} \sin(\theta_1 - \theta_o(x_1)) \frac{d\theta_o}{dx_1} \end{aligned} \quad (163) $$

As discussed in the Appendix and mentioned in the introduction to the $\hat{\chi}_\pm$ operator, spin is responsible for keeping $\hat{x}$ and $\hat{x}_o$ aligned so that a particle propagates in the same direction in the absence of an external force. It is thus postulated, in line with spin-orbit coupling, that the phase vector component $k_s$ is determined by the following relationship:

$$ s\hbar = \hbar k_s a \quad (164) $$

<figure><img src="image_9.png" /><figcaption>FIG. 8. Evolving waveform of two-slit interference for transverse wave propagation. Here, the modified Huygen's principle can be used to determine the phase at the evaluation point of the quanta, which obtains the same result as determined from standard theoretical analysis for locations far from the slits. The sketch is only illustrative, stretched out over time, as it does not restrict the waveform to only the phase bubble region in the vicinity of the quanta. Close to the slits, (166) specifies that the carrier rotates about the quanta (center) after it passes through the left slit. Considering two-dimensional analysis of the waveform within the grating, a small tunneling probability is present that would allow the quanta to tunnel between slits, but the occurrence of this is not necessary to characterize two-slit interference.</figcaption></figure>

where the slit separation of $a$ is the moment arm of the effective angular momentum associated with the carrier phase rotation. The phase angle then becomes:

$$ \theta_o(x_1) = \frac{\pi}{2} + k_s x_1 = \frac{\pi}{2} + \frac{s}{a} x_1 \quad (165) $$

This gives the final equation that determines $\alpha_\parallel$:

$$ \alpha_\parallel^{-1} \approx \alpha_o^{-1} + \frac{s\lambda_a}{4a} \sin(\theta_1 - \theta_o(x_1)) \Theta(\theta_1 - \theta_o(x_1)) \quad (166) $$

where $\Theta$ is the Heaviside function. Thus, the carrier propagation direction is in alignment with the quanta's motion when:

$$ x_1 \gtrsim \frac{a \left| \theta_1 - \frac{\pi}{2} \right|}{s} \quad (167) $$

At this distance, the calculation of the probability in (149) is becoming closer to the correct value.

For the rest of this paper, additional well-known problems will be analyzed with this theory. Explicit consideration of $\hat{\chi}_\pm$ will be dropped in the next few sections to simplify the derivations as phase offsets are determined by boundary conditions.
VIII. FINITE SQUARE WELL

Setting $\phi_o = 0$, $\frac{\pi}{2}$ in (43) produces an orthogonal set $sin^2(k_e x)$, $cos^2(k_e x)$ for $\nu$ with the same assignments for $P(x)$ in (21), only switched for the respective values of $\phi_o$, satisfying the symmetry requirement within a centered square well with boundaries at $\pm \frac{L}{2}$.

For $E - m_o c^2 < V_o$, the phase projection of (7) can temporarily lower the barrier for freshly injected particles that have not yet mediated their phase. The barrier lowering may not be enough, which causes reflections leading to a trap for the particle, or if the barrier is effectively lowered, the phase projection term becomes constant and it is up to the virtual forces inside the barrier to either reflect the particle or permit it to travel deeper into the barrier. As discussed, this depends upon the spin state.

The reason the barrier phase projection term is constant is that the quantum state is no longer harmonic in this region and its phase, including phase noise, is pinned at its entry value into the barrier. The average of the virtual force operator $\hat{h}_s \cdot \hat{\chi}_\pm$ is a different constant value than before, depending on the direction of the spin and the motion of the particle. As such, the work calculation (33) takes a simpler form, but there should be a continuity to (40) for a particle that travels above and then below the barrier.

This continuity requirement forces a change to occur for $\nu_x$. Unlike the prior free particle case, below the barrier:

$$ \nu_x = \nu(x) \qquad (168) $$

i.e. the particle's mediated oscillation frequency $\nu(x)$ dictates the virtual force's exchange frequency $\nu_x$ in steady-state equilibrium.

Using this in (36) and (9) with differences in the virtual work derivation accounted for in $\nu(x)$, gives:

$$ \frac{d\lambda}{dx} = \pm 2\pi \qquad (169) $$

with solution:

$$ \lambda_{\pm} = \lambda' \pm 2\pi d \qquad (170) $$

where $d = |x| - \frac{L}{2}$ is the penetration depth into a barrier and, due to the expression of negative kinetic energy $E - m_o c^2 - V_o$, the sign of $\lambda'$ is absorbed into the phase such that it is in line with Schrödinger's non-relativistic characterization of the effective decay constant inside the barrier:

$$ \frac{2\pi}{\lambda'} \equiv \sqrt{\frac{2m_o [V_o - (E - m_o c^2)]}{\hbar^2}} \qquad (171) $$

Traditionally, in calculating $\lambda'$, E represents kinetic energy, however in this treatment the dynamics of quantum mass is coupled to $\lambda$ and $\nu$ and so it is considered a part of total energy E. In the next section, a different assumption for $\lambda'$ is used for a barrier grown on a semiconductor, with charges free to adjust the effective barrier height.

As $E - m_o c^2 < V_o$, the reservoir of exchange energy $h\nu_x$ resides with $V_o$, hence $\lambda_F$ is constant. This together with (168) and (36) specifies that $\nu$ varies with position in tandem with $\lambda_\pm$:

$$ \frac{\lambda_{\pm}}{\nu} = \text{constant} \qquad (172) $$

As $\nu$, not $\lambda$, is continuous across a well boundary, one obtains the following from (16) and (17) for $P_\pm(x)$ inside the barrier:

$$ P_{\pm}(x) = N_{\pm}e^{-4\pi \int \frac{dx'}{\lambda' \pm 2\pi(x' - \frac{L}{2})}} = N\lambda_a^{\pm 2} [\lambda' \pm 2\pi d]^{\mp 2} \qquad (173) $$

To solve (172) for $\nu(x)$, first (43) is used to determine the frequency at the right barrier boundary ($x = \frac{L}{2}$), then it is multiplied by the normalized wavelength from (170):

$$ \nu_{\pm} = \nu_a \sin^2 \left( \frac{k_e L}{2} - \phi_o \right) \left( 1 \pm \frac{2\pi d}{\lambda'} \right) \qquad (174) $$

In the non-relativistic limit, $\gamma = 1$ and the energy inside the barrier is not expressed with harmonic terms as was done for $E - m_o c^2 > V_o$. Instead, the summation of quantum mass-energy and the energy of oscillation is equivalent to the gap between a particle's energy and the potential, which yields the expression:

$$ \begin{aligned} m_{\pm}c^2 &= E - h\nu_{\pm} - V_o && (175) \\ &= h\nu_a \left[ 1 - \sin^2 \left( \frac{k_e L}{2} - \phi_o \right) \left( 1 \pm \frac{2\pi d}{\lambda'} \right) \right] - h\nu' \end{aligned} $$

where $m_+ \to 0$ with an increase in $\nu_+$ leads to saturation. Figure 9 illustrates the overall solution to this problem which is broken into two parts: a quasi-equilibrium probability distribution and the quasi steady-state distribution. The quasi-equilibrium solution consists of those particles that return to or stay within the well. To determine which solutions are consistent with this, as considered further in Appendix A, one can infer that there is a unique spin state corresponding to the $\lambda_+$ and $\lambda_-$ solutions inside the barrier region.

I.e., hidden in this 1-D derivation is the virtual force analysis carried out previously with a constant potential. The virtual force $\mathbf{F} = F\hat{\mathbf{h}}_s$ inside the barrier points along the direction of the helicity $\hat{\mathbf{h}}_s$ while conservation applies to the spin's component in a plane orthogonal to $\mathbf{F}$. For the $\lambda_+$ state, $\hat{\mathbf{h}}_s$ points in the direction of motion, hence $\mathbf{F}$ speeds the particle up inside the barrier, while for the $\lambda_-$ state, $\hat{\mathbf{h}}_s$ points in the opposite direction of motion, hence the particle slows down until either phase inversion returns it to the well due to phase noise or the probability node is reached and $\hat{\mathbf{h}}_s$ reverses at $d = \frac{\lambda'}{2\pi}$.

Only the $\lambda_-$ solution is shown for the quasi-equilibrium solution because the virtual force makes reflection possible. The equilibrium solution is not a true equilibrium because a certain percentage of these particles will exchange with others assigned to the quasi steady-state distribution, which leads to their escape for a finite barrier width.
VIII. FINITE SQUARE WELL

Setting $\phi_o = 0$, $\frac{\pi}{2}$ in (43) produces an orthogonal set $sin^2(k_e x)$, $cos^2(k_e x)$ for $\nu$ with the same assignments for $P(x)$ in (21), only switched for the respective values of $\phi_o$, satisfying the symmetry requirement within a centered square well with boundaries at $\pm \frac{L}{2}$.

For $E - m_o c^2 < V_o$, the phase projection of (7) can temporarily lower the barrier for freshly injected particles that have not yet mediated their phase. The barrier lowering may not be enough, which causes reflections leading to a trap for the particle, or if the barrier is effectively lowered, the phase projection term becomes constant and it is up to the virtual forces inside the barrier to either reflect the particle or permit it to travel deeper into the barrier. As discussed, this depends upon the spin state.

The reason the barrier phase projection term is constant is that the quantum state is no longer harmonic in this region and its phase, including phase noise, is pinned at its entry value into the barrier. The average of the virtual force operator $\hat{h}_s \cdot \hat{\chi}_\pm$ is a different constant value than before, depending on the direction of the spin and the motion of the particle. As such, the work calculation (33) takes a simpler form, but there should be a continuity to (40) for a particle that travels above and then below the barrier.

This continuity requirement forces a change to occur for $\nu_x$. Unlike the prior free particle case, below the barrier:

$$ \nu_x = \nu(x) \qquad (168) $$

i.e. the particle's mediated oscillation frequency $\nu(x)$ dictates the virtual force's exchange frequency $\nu_x$ in steady-state equilibrium.

Using this in (36) and (9) with differences in the virtual work derivation accounted for in $\nu(x)$, gives:

$$ \frac{d\lambda}{dx} = \pm 2\pi \qquad (169) $$

with solution:

$$ \lambda_{\pm} = \lambda' \pm 2\pi d \qquad (170) $$

where $d = |x| - \frac{L}{2}$ is the penetration depth into a barrier and, due to the expression of negative kinetic energy $E - m_o c^2 - V_o$, the sign of $\lambda'$ is absorbed into the phase such that it is in line with Schrödinger's non-relativistic characterization of the effective decay constant inside the barrier:

$$ \frac{2\pi}{\lambda'} \equiv \sqrt{\frac{2m_o [V_o - (E - m_o c^2)]}{\hbar^2}} \qquad (171) $$

Traditionally, in calculating $\lambda'$, E represents kinetic energy, however in this treatment the dynamics of quantum mass is coupled to $\lambda$ and $\nu$ and so it is considered a part of total energy E. In the next section, a different assumption for $\lambda'$ is used for a barrier grown on a semiconductor, with charges free to adjust the effective barrier height.

As $E - m_o c^2 < V_o$, the reservoir of exchange energy $h\nu_x$ resides with $V_o$, hence $\lambda_F$ is constant. This together with (168) and (36) specifies that $\nu$ varies with position in tandem with $\lambda_\pm$:

$$ \frac{\lambda_{\pm}}{\nu} = \text{constant} \qquad (172) $$

As $\nu$, not $\lambda$, is continuous across a well boundary, one obtains the following from (16) and (17) for $P_\pm(x)$ inside the barrier:

$$ P_{\pm}(x) = N_{\pm}e^{-4\pi \int \frac{dx'}{\lambda' \pm 2\pi(x' - \frac{L}{2})}} = N\lambda_{\pm}^{2}[\lambda' \pm 2\pi d]^{\mp 2} \qquad (173) $$

To solve (172) for $\nu(x)$, first (43) is used to determine the frequency at the right barrier boundary ($x = \frac{L}{2}$), then it is multiplied by the normalized wavelength from (170):

$$ \nu_{\pm} = \nu_a \sin^2 \left( \frac{k_e L}{2} - \phi_o \right) \left( 1 \pm \frac{2\pi d}{\lambda'} \right) \qquad (174) $$

In the non-relativistic limit, $\gamma = 1$ and the energy inside the barrier is not expressed with harmonic terms as was done for $E - m_o c^2 > V_o$. Instead, the summation of quantum mass-energy and the energy of oscillation is equivalent to the gap between a particle's energy and the potential, which yields the expression:

$$ \begin{aligned} m_{\pm}c^2 &= E - h\nu_{\pm} - V_o && (175) \\ &= h\nu_a \left[ 1 - \sin^2 \left( \frac{k_e L}{2} - \phi_o \right) \left( 1 \pm \frac{2\pi d}{\lambda'} \right) \right] - h\nu' \end{aligned} $$

where $m_+ \to 0$ with an increase in $\nu_+$ leads to saturation. Figure 9 illustrates the overall solution to this problem which is broken into two parts: a quasi-equilibrium probability distribution and the quasi steady-state distribution. The quasi-equilibrium solution consists of those particles that return to or stay within the well. To determine which solutions are consistent with this, as considered further in Appendix A, one can infer that there is a unique spin state corresponding to the $\lambda_+$ and $\lambda_-$ solutions inside the barrier region.

I.e., hidden in this 1-D derivation is the virtual force analysis carried out previously with a constant potential. The virtual force $\mathbf{F} = F\hat{\mathbf{h}}_s$ inside the barrier points along the direction of the helicity $\hat{\mathbf{h}}_s$ while conservation applies to the spin's component in a plane orthogonal to $\mathbf{F}$. For the $\lambda_+$ state, $\hat{\mathbf{h}}_s$ points in the direction of motion, hence $\mathbf{F}$ speeds the particle up inside the barrier, while for the $\lambda_-$ state, $\hat{\mathbf{h}}_s$ points in the opposite direction of motion, hence the particle slows down until either phase inversion returns it to the well due to phase noise or the probability node is reached and $\hat{\mathbf{h}}_s$ reverses at $d = \frac{\lambda'}{2\pi}$.

Only the $\lambda_-$ solution is shown for the quasi-equilibrium solution because the virtual force makes reflection possible. The equilibrium solution is not a true equilibrium because a certain percentage of these particles will exchange with others assigned to the quasi steady-state distribution, which leads to their escape for a finite barrier width.
<figure><img src="image_11.png" /><figcaption>FIG. 9. Square well with a finite barrier thickness. The solutions shown are from quasi-equilibrium and quasi-steady state analysis. Barrier adjustment due to the concentration of charge is exaggerated for illustration purposes only. The case illustrated is from a study [28] discussed in the next section where the probability node adjusts such that $\lambda_-$ current becomes saturated at the barrier's opposing effective boundary.</figcaption></figure>

Furthermore, the analysis carried out here is approximate in terms of its characterization of barrier alteration due to image charges or concentrations of electrons inside the barrier. The assumption used here is that the concentration of charge near the edges of the barrier reduces the overall barrier height and effective width. As depicted in the illustration, the free-particle behavior of the $\lambda_-$ electrons traveling beyond the effective width of the barrier corresponds to the case where current saturation is observed in the experiment discussed in the next section.

While (173) does not agree with solutions to Schrödinger's equation inside the barrier region due to the virtual force concept, continuity requirements for $|\phi\rangle = \sqrt{P}$ and $\frac{d\phi}{dx}$ at the boundary, $|x| - \frac{L}{2}$, with $|\phi\rangle \propto \cos(k_e x - \phi_o)$ for the well and $|\phi\rangle \propto [\lambda' \pm 2\pi(x - \frac{L}{2})]^{\mp 1}$ for the barrier region, generate the same energy quantization values for the even and odd states:

$$k_e \tan \left( \frac{k_e L}{2} - \phi_o \right) = \frac{2\pi}{\lambda'} \quad (176)$$

where $\phi_o = 0$, $\frac{\pi}{2}$ specifies even or odd states, respectively, and $\lambda'$ takes its value from (171).

With the mass dynamics inside different than outside a barrier, a speedup consistent with the finding in [3] occurs for $\lambda_+$ states that is not superluminal. A measurement of $\lambda_-$ states on the other hand, absent from the solution to Schrödinger's equation, would see a slowdown for very thin barriers.

## IX. TUNNELING CURRENT

A metal/oxide junction's [28] step potential is modeled by a square well with $x = 0$ referenced to the Fermi level at distance $\frac{L}{2}$ away from the barrier. Referring to (171), as $\lambda'$'s discontinuity across the barrier creates a

discontinuity in charge density, the barrier adjusts such that $\lambda$ is continuous, i.e. $\lambda' = \lambda(\frac{L}{2})$.

As presented in this work, the evolution of the carrier and envelope are coupled, however they can be analyzed separately. In addition, as long as the wavelengths and frequencies are paired correctly, the dynamics of a particle can be characterized either by the amplitudes of the carrier or the envelope since they travel at the same velocity. For this problem, the wavelength amplitude of the electron is used.

The Fermi level $E_f$ rises continuously, represented by the even solution from (25), $\lambda = \lambda_a \cot(k_f x - \phi_b)$, where $\phi_b$, affected by barrier adjustment, sets a finite value for $\lambda$ when the Fermi wavevector $k_f = 0$ at $V = 0$. Defining:

$$u \equiv \cot \left( \frac{k_f L}{2} - \phi_b \right) = \cot(\pi L \sqrt{\Phi - \Phi_b}) \quad (177)$$

where $\Phi \equiv \beta V$, $\beta = \frac{2m^*q}{\hbar^2}$ assumes an effective mass $m^*$, and:

$$E_f \equiv h\nu_e = qV = \frac{\hbar^2}{2m^*\lambda_e^2} = \frac{\hbar^2 k_f^2}{2m^*} = \frac{\hbar^2}{2m^*\lambda_a^2} - E_o \quad (178)$$

where reference energy $E_o$ is related to $\Phi_o \equiv \frac{\beta E_o}{q}$ by the following:

$$\frac{\beta}{q}(E_f + E_o) = \frac{1}{\lambda_e^2} + \Phi_o = \frac{1}{\lambda_a^2} \equiv \Phi + \Phi_o \quad (179)$$

Using (170), (173), and (174), and noting $\lambda' = \lambda_a u$ along with $\nu_a = \alpha_\parallel \nu_e$, where (179) suggests $\alpha_\parallel$, the ratio of $\lambda_e$ to $\lambda_a$, is constant, the dominant, Fermi-level contributions to the current densities $J_\pm$ are:

$$\mathbf{J}_+ = qP_+(d)\lambda_+(d)\nu_+(d) = \frac{J_o}{u} \left\{ \Phi_{sat}^+ \approx \frac{cot^6(\pi L \sqrt{\Phi_{sat}^+ - \Phi_b})}{(2\pi d)^2} \left(1 - \frac{\nu'}{\nu_a}\right)^2 - \Phi_o \right\} \quad (180)$$

$$\mathbf{J}_- = qP_-(d)\lambda_-(d)\nu_-(d) = \frac{J_o}{u} \left( u - \frac{2\pi d}{\lambda_a} \right)^4 \\ = \mathbf{J}_+(u - 2\pi d \sqrt{\Phi + \Phi_o})^4 \\ \left\{ \Phi_{sat}^- \approx \frac{cot^2(\pi L \sqrt{\Phi_{sat}^- - \Phi_b})}{(\pi d)^2} - \Phi_o \right\} \\ J_o \equiv qN\alpha_\parallel\lambda_a\nu_e\sin^2\left(\frac{k_f L}{2}\right) = \frac{q^2 N \alpha_\parallel \sin^2(\pi L \sqrt{\Phi})}{h \sqrt{\Phi + \Phi_o}} V \quad (181)$$

Consistent with the well-known $V^2$ dependence at low bias, these currents reach steady-state values with the phase of electrons adjusting as they enter the device, allowing some to penetrate and travel through the barrier. The approximate saturation value of $\Phi$ shown for $J_+$ can
<figure><img src="image_10.png" /><figcaption>FIG. 9. Square well with a finite barrier thickness. The solutions shown are from quasi-equilibrium and quasi-steady state analysis. Barrier adjustment due to the concentration of charge is exaggerated for illustration purposes only. The case illustrated is from a study [28] discussed in the next section where the probability node adjusts such that $\lambda_-$ current becomes saturated at the barrier's opposing effective boundary.</figcaption></figure>

Furthermore, the analysis carried out here is approximate in terms of its characterization of barrier alteration due to image charges or concentrations of electrons inside the barrier. The assumption used here is that the concentration of charge near the edges of the barrier reduces the overall barrier height and effective width. As depicted in the illustration, the free-particle behavior of the $\lambda_-$ electrons traveling beyond the effective width of the barrier corresponds to the case where current saturation is observed in the experiment discussed in the next section.

While (173) does not agree with solutions to Schrödinger's equation inside the barrier region due to the virtual force concept, continuity requirements for $|\phi\rangle = \sqrt{P}$ and $\frac{d\phi}{dx}$ at the boundary, $|x| - \frac{L}{2}$, with $|\phi\rangle \propto \cos(k_e x - \phi_o)$ for the well and $|\phi\rangle \propto [\lambda' \pm 2\pi(x - \frac{L}{2})]^{\mp 1}$ for the barrier region, generate the same energy quantization values for the even and odd states:

$$k_e \tan \left( \frac{k_e L}{2} - \phi_o \right) = \frac{2\pi}{\lambda'} \quad (176)$$

where $\phi_o = 0$, $\frac{\pi}{2}$ specifies even or odd states, respectively, and $\lambda'$ takes its value from (171).

With the mass dynamics inside different than outside a barrier, a speedup consistent with the finding in [3] occurs for $\lambda_+$ states that is not superluminal. A measurement of $\lambda_-$ states on the other hand, absent from the solution to Schrödinger's equation, would see a slowdown for very thin barriers.

## IX. TUNNELING CURRENT

A metal/oxide junction's [28] step potential is modeled by a square well with $x = 0$ referenced to the Fermi level at distance $\frac{L}{2}$ away from the barrier. Referring to (171), as $\lambda'$'s discontinuity across the barrier creates a

discontinuity in charge density, the barrier adjusts such that $\lambda$ is continuous, i.e. $\lambda' = \lambda(\frac{L}{2})$.

As presented in this work, the evolution of the carrier and envelope are coupled, however they can be analyzed separately. In addition, as long as the wavelengths and frequencies are paired correctly, the dynamics of a particle can be characterized either by the amplitudes of the carrier or the envelope since they travel at the same velocity. For this problem, the wavelength amplitude of the electron is used.

The Fermi level $E_f$ rises continuously, represented by the even solution from (25), $\lambda = \lambda_a \cot(k_f x - \phi_b)$, where $\phi_b$, affected by barrier adjustment, sets a finite value for $\lambda$ when the Fermi wavevector $k_f = 0$ at $V = 0$. Defining:

$$u \equiv \cot \left( \frac{k_f L}{2} - \phi_b \right) = \cot(\pi L \sqrt{\Phi - \Phi_b}) \quad (177)$$

where $\Phi \equiv \beta V$, $\beta = \frac{2m^* q}{\hbar^2}$ assumes an effective mass $m^*$, and:

$$E_f \equiv h\nu_e = qV = \frac{\hbar^2}{2m^*\lambda_e^2} = \frac{\hbar^2 k_f^2}{2m^*} = \frac{\hbar^2}{2m^*\lambda_a^2} - E_o \quad (178)$$

where reference energy $E_o$ is related to $\Phi_o \equiv \frac{\beta E_o}{q}$ by the following:

$$\frac{\beta}{q}(E_f + E_o) = \frac{1}{\lambda_e^2} + \Phi_o = \frac{1}{\lambda_a^2} \equiv \Phi + \Phi_o \quad (179)$$

Using (170), (173), and (174), and noting $\lambda' = \lambda_a u$ along with $\nu_a = \alpha_\parallel \nu_e$, where (179) suggests $\alpha_\parallel$, the ratio of $\lambda_e$ to $\lambda_a$, is constant, the dominant, Fermi-level contributions to the current densities $J_\pm$ are:

$$\mathbf{J}_+ = qP_+(d)\lambda_+(d)\nu_+(d) = \frac{J_o}{u} \left\{ \Phi_{sat}^+ \approx \frac{cot^6(\pi L \sqrt{\Phi_{sat}^+ - \Phi_b})}{(2\pi d)^2} \left(1 - \frac{\nu'}{\nu_a}\right)^2 - \Phi_o \right\} \quad (180)$$

$$\mathbf{J}_- = qP_-(d)\lambda_-(d)\nu_-(d) = \frac{J_o}{u} \left( u - \frac{2\pi d}{\lambda_a} \right)^4 \\ = \mathbf{J}_+(u - 2\pi d \sqrt{\Phi + \Phi_o})^4 \\ \left\{ \Phi_{sat}^- \approx \frac{cot^2(\pi L \sqrt{\Phi_{sat}^- - \Phi_b})}{(\pi d)^2} - \Phi_o \right\} \\ J_o \equiv qN\alpha_\parallel\lambda_a\nu_e \sin^2\left(\frac{k_f L}{2}\right) = \frac{q^2 N \alpha_\parallel \sin^2(\pi L \sqrt{\Phi})}{h \sqrt{\Phi + \Phi_o}} V \quad (181)$$

Consistent with the well-known $V^2$ dependence at low bias, these currents reach steady-state values with the phase of electrons adjusting as they enter the device, allowing some to penetrate and travel through the barrier. The approximate saturation value of $\Phi$ shown for $J_+$ can
be found using (175) for $m_+ = 0$. This gives the following equation to solve with $k_f \equiv k_e$:

$$ \left( 1 + \frac{2\pi d \sqrt{\Phi_{sat}^+ + \Phi_o}}{u} \right) \approx (1 + u^2) \left( 1 - \frac{\nu'}{\nu_a} \right) \quad (182) $$

utilizing $\sin^2(\frac{k_f L}{2} - \phi_o) \approx (1 + u^2)^{-1}$ and the approximation of (179). Given $\frac{\nu'}{\nu_a} \ll 1$, terms cancel to yield the following simplified equation:

$$ \sqrt{\Phi_{sat}^+ + \Phi_o} \approx \frac{u^3}{2\pi d} \left( 1 - \frac{\nu'}{\nu_a} \right) \quad (183) $$

which reduces to the expression shown in (180) assuming parabolic bands.

At the probability node, $d = \frac{u\lambda_a}{2\pi}$, both $\nu$ and $\lambda_-$ vanish. Due to phase noise, this is a point of instability, and while some electrons may reflect back, others continue on. As to what it means to have a probability node, according to (29), there is no virtual force at that location which implies $\langle\phi'_F|\phi\rangle = 0$, i.e. the particle is not interacting with the background, and is undetectable.

In general, this is expected to be true: probability nodes are points of instability and particles pass through them undetected, unless there is phase noise. Given the symmetrical $\lambda$, $\nu$ profile about the node in the barrier, which depends on bias, the conservation of $J_-$ electrons and flow past it can't exceed the maximum near the well boundary. This specifies $\Phi_{sat}^-$, i.e. the point at which $J_-$ reaches the same peak value on the other side of the node, at which point the electrons have left the barrier and the spin component along the direction of motion and virtual force settle into an oscillatory pattern with zero net work, as discussed previously for the virtual force associated with a constant potential.

Figure 10 shows two features unmodeled in [28]: a kink in the conductance at $V = .17eV$, attributed to inelastic phonon scattering by the authors [28], and a cusp around $V = 0$, attributed to a Coulomb blockade. Even with the consideration of a single dominant phonon energy in disordered material [29], there has been no discontinuity in the derivative of the flow seen in the experiments of other studies, and given the reproducibility of the same kink across devices, with the standard modeling of scattering processes both elastic and inelastic [30], the variances in the depth of flow with energy do not generate sharp inflections in the current density. Regarding the blockade, it has been shown [31] that capacitive and inductive modeling of the junction doesn't reproduce the inflection seen in the data whereby the conductance rises then falls only to rise again.

With this study, the saturation of $J_-$ acts as a leakage current leading to the kink whereas the cusp is caused by the rapidly changing, initially large wavelength near zero applied bias whereby the capacitive effects of the junction are modeled by the barrier adjustment introduced with $\Phi_b$. The fit was generated using the measured effective mass $m^* = 0.4m_o$ of $Al_2O_3$, $\frac{\Phi_b}{\beta} = -0.037V$,

<figure><img src="image_12.png" /><figcaption>FIG. 10. Replicating the observed tunnel current conductance with two unique states inside the barrier region. Absent from the original paper's analysis [28], the cusp near the origin is due to the dramatic response of the wavelength near zero bias and the kink at .17eV is due to the saturation of $J_-$ current in reaction to a probability node at depth $\frac{\lambda_-}{2\pi}$ not present in the exponential decay solution to Schrödinger's equation.</figcaption></figure>

$\frac{\Phi_o}{\beta} = -0.06V$, $L = 10.6$, and $d = 11.2$ within 10% of the stated value the authors determined from fits to the Simmons model [32]. Though not considered here, the slight asymmetry shown in [28] for the conductance at reverse bias of the two electrodes is readily modeled by different values for $\Phi_b$ and L of the other electrode.

The next standard approximation for the analysis of tunnel current models the potential $\Phi$ applied across a barrier of thickness $d$ as a trapezoid:

$$ V = H - \frac{q\Phi}{d} \left( x - \frac{L}{2} \right) \quad (184) $$

In the non-relativistic limit, with kinetic energy $T = \frac{1}{2}m\lambda_e^2\nu_e^2 = \frac{1}{2}h\nu_e$, a different approximation to the relationship between $\nu_e$ and energy using the effective mass is:

$$ E \approx \frac{1}{2}h\nu_e(x) + m^*c^2 - \frac{q\Phi}{d}\left(x - \frac{L}{2}\right) \quad (185) $$

Here, the barrier height $H$ is not a part of the expression since on average over all incident phases $\phi$, no work is done during barrier penetration by the barrier boundary's electric field. The applied electrostatic field is not confined within an infinitely small width, however, so work is done and its impact is included.

Given similar considerations for tunneling as for a constant applied force analyzed in Section IV, the phase projection term associated with the constant applied force is +1 due to the absence of a harmonic phase for the tunneling states. Hence, assuming steady-state equilibrium, (9) can be applied with $\frac{h\nu'}{\lambda_F} = -\frac{q\Phi}{d}$, giving the following equation:

$$ \frac{d\lambda_{\pm}}{dx} = - \frac{\frac{q\Phi}{2d}\lambda_{\pm}}{(E_f - \frac{q\Phi}{d}\frac{L}{2}) + \frac{q\Phi}{d}x} \pm 2\pi \quad (186) $$

where $F$ assigned to $H$ is virtual, contributing the factor of $\pm 2\pi$, and $E_f$ represents the Fermi level energy $E$ -
be found using (175) for $m_+ = 0$. This gives the following equation to solve with $k_f \equiv k_e$:

$$ \left( 1 + \frac{2\pi d \sqrt{\Phi_{sat}^+ + \Phi_o}}{u} \right) \approx (1 + u^2) \left( 1 - \frac{\nu'}{\nu_a} \right) \quad (182) $$

utilizing $\sin^2(\frac{k_f L}{2} - \phi_o) \approx (1 + u^2)^{-1}$ and the approximation of (179). Given $\frac{\nu'}{\nu_a} \ll 1$, terms cancel to yield the following simplified equation:

$$ \sqrt{\Phi_{sat}^{+} + \Phi_o} \approx \frac{u^3}{2\pi d} \left( 1 - \frac{\nu'}{\nu_a} \right) \quad (183) $$

which reduces to the expression shown in (180) assuming parabolic bands.

At the probability node, $d = \frac{u\lambda_a}{2\pi}$, both $\nu$ and $\lambda_-$ vanish. Due to phase noise, this is a point of instability, and while some electrons may reflect back, others continue on. As to what it means to have a probability node, according to (29), there is no virtual force at that location which implies $\langle\phi'_F|\phi\rangle = 0$, i.e. the particle is not interacting with the background, and is undetectable.

In general, this is expected to be true: probability nodes are points of instability and particles pass through them undetected, unless there is phase noise. Given the symmetrical $\lambda$, $\nu$ profile about the node in the barrier, which depends on bias, the conservation of $J_-$ electrons and flow past it can't exceed the maximum near the well boundary. This specifies $\Phi_{sat}^-$, i.e. the point at which $J_-$ reaches the same peak value on the other side of the node, at which point the electrons have left the barrier and the spin component along the direction of motion and virtual force settle into an oscillatory pattern with zero net work, as discussed previously for the virtual force associated with a constant potential.

Figure 10 shows two features unmodeled in [28]: a kink in the conductance at $V = .17eV$, attributed to inelastic phonon scattering by the authors [28], and a cusp around $V = 0$, attributed to a Coulomb blockade. Even with the consideration of a single dominant phonon energy in disordered material [29], there has been no discontinuity in the derivative of the flow seen in the experiments of other studies, and given the reproducibility of the same kink across devices, with the standard modeling of scattering processes both elastic and inelastic [30], the variances in the depth of flow with energy do not generate sharp inflections in the current density. Regarding the blockade, it has been shown [31] that capacitive and inductive modeling of the junction doesn't reproduce the inflection seen in the data whereby the conductance rises then falls only to rise again.

With this study, the saturation of $J_-$ acts as a leakage current leading to the kink whereas the cusp is caused by the rapidly changing, initially large wavelength near zero applied bias whereby the capacitive effects of the junction are modeled by the barrier adjustment introduced with $\Phi_b$. The fit was generated using the measured effective mass $m^* = 0.4m_o$ of $Al_2O_3$, $\frac{\Phi_b}{\beta} = -0.037V$,

<figure><img src="image_11.png" /><figcaption>FIG. 10. Replicating the observed tunnel current conductance with two unique states inside the barrier region. Absent from the original paper's analysis [28], the cusp near the origin is due to the dramatic response of the wavelength near zero bias and the kink at .17eV is due to the saturation of $J_-$ current in reaction to a probability node at depth $\frac{\lambda}{2\pi}$ not present in the exponential decay solution to Schrödinger's equation.</figcaption></figure>

$\frac{\Phi_o}{\beta} = -0.06V$, $L = 10.6$, and $d = 11.2$ within 10% of the stated value the authors determined from fits to the Simmons model [32]. Though not considered here, the slight asymmetry shown in [28] for the conductance at reverse bias of the two electrodes is readily modeled by different values for $\Phi_b$ and L of the other electrode.

The next standard approximation for the analysis of tunnel current models the potential $\Phi$ applied across a barrier of thickness $d$ as a trapezoid:

$$ V = H - \frac{q\Phi}{d} \left( x - \frac{L}{2} \right) \quad (184) $$

In the non-relativistic limit, with kinetic energy $T = \frac{1}{2}m\lambda_e^2\nu_e^2 = \frac{1}{2}h\nu_e$, a different approximation to the relationship between $\nu_e$ and energy using the effective mass is:

$$ E \approx \frac{1}{2}h\nu_e(x) + m^*c^2 - \frac{q\Phi}{d}\left(x - \frac{L}{2}\right) \quad (185) $$

Here, the barrier height $H$ is not a part of the expression since on average over all incident phases $\phi$, no work is done during barrier penetration by the barrier boundary's electric field. The applied electrostatic field is not confined within an infinitely small width, however, so work is done and its impact is included.

Given similar considerations for tunneling as for a constant applied force analyzed in Section IV, the phase projection term associated with the constant applied force is +1 due to the absence of a harmonic phase for the tunneling states. Hence, assuming steady-state equilibrium, (9) can be applied with $\frac{h\nu'}{\lambda_F} = -\frac{q\Phi}{d}$, giving the following equation:

$$ \frac{d\lambda_{\pm}}{dx} = - \frac{\frac{q\Phi}{2d}\lambda_{\pm}}{(E_f - \frac{q\Phi}{d}\frac{L}{2}) + \frac{q\Phi}{d}x} \pm 2\pi \quad (186) $$

where $F$ assigned to $H$ is virtual, contributing the factor of $\pm 2\pi$, and $E_f$ represents the Fermi level energy $E$ -
$m^*c^2$. Using the same boundary condition $\lambda' = \lambda_{\pm}(\frac{L}{2})$ as before:

$$ \lambda_{\pm}(x) = \pm \frac{4\pi}{3} \left( \frac{E_f d}{q\Phi} + x - \frac{L}{2} \right) + \left( u\lambda_a \mp \frac{4\pi}{3} \frac{E_f d}{q\Phi} \right) \left[ 1 + \frac{q\Phi}{E_f} \frac{x - \frac{L}{2}}{d} \right]^{-\frac{1}{2}} \quad (187) $$

While a full analysis can be performed, the focus here is on a comparison to the exponential probability factors of other approaches such as WKB. Assuming $u\lambda_a \gg \frac{E_f d}{q\Phi}$ or considering high voltages or electric fields, $\frac{E_f}{q\Phi} \ll 1$, to good approximation with the expansion of the square root:

$$ \lambda_{\pm} \approx u\lambda_a \left[ 1 + \frac{q\Phi}{E_f} \frac{x - \frac{L}{2}}{d} \right]^{-\frac{1}{2}} \pm 2\pi \left( x - \frac{L}{2} \right) \quad (188) $$

Ignoring the second term for the moment in considering thin barriers and substituting (188) into (17), the solution takes the following form:

$$ P_+(x=d) \approx N e^{-\frac{8\pi}{3u} \frac{d}{q\hbar\Phi} (2m^*)^{\frac{1}{2}} [E_f + q\Phi]^{\frac{3}{2}}} \\ = N e^{-\frac{8\pi}{3u} \frac{d}{q\hbar\Phi} (2m^*)^{\frac{1}{2}} [\frac{1}{2}h\nu_e]^{\frac{3}{2}}} \quad (189) $$

where the presence of $\nu_e$ is the result of using (185) at thickness $d$, assuming the $\lambda_{\pm}$ solutions combine to give an effective $\nu_e$.

This formula matches the exponential of the well-known tunneling current analysis of Fowler-Nordheim [33] and Simmons [32] (2.96 replacing 3) for high voltage, if one removes the quantum well, thus setting $u$ to 1 and $H - E_f$ takes the place of $\frac{1}{2}h\nu_e$. What is not shown is how the wavelength and frequency change to bolster the velocity of the carriers as the exponential changes with applied bias. Depending on $V$ and $d$, $\nu_{e-}$ could exponentially increase beyond the node, overshooting saturation to the point that it leads to a premature breakdown of the oxide.

Tunneling current as a function of barrier thickness has been compared in one study [34] to calculations with the Simmons model that predict thicknesses roughly 2.2 times smaller than optical measurements of barriers 20Å thick with an electric field on the order of $15 \times 10^6 \frac{V}{cm}$ (30% from breakdown), and other studies have shown such discrepancies cannot be attributed to defects [35]. Disagreement between the model and experiment tapers slightly to a factor of 2 at 32 Å oxide thickness.

To investigate how this theory's calculation would fair, it first has to be recognized that standard formulas predict an increase in tunneling current with increasing kinetic energy of the carrier. At first, this theory appears to predict the opposite because barrier penetration is more likely for lower frequency components, i.e. longer wavelengths that involve slower evolving phases that permit deeper penetration. However, in this derivation with a

quantum well, what matters is the normalization that is associated with this dependency. The normalization is a function of $u$, which in turn is dependent on the solution to (176). Hence, while the kinetic energy may increase, thus creating a more negative exponent in (189) according to the approximations introduced, $u$ can increase (which is typically seen in finite barrier solutions), thereby reducing the magnitude of the exponent. If it is assumed that the Fermi level is pinned by the bulk, unlike this case study where it was assumed to be affected by adjustment at the barrier, then the current's dependency on $u$ and $\Phi$ or $V$ is clearly dominant.

One can also characterize $\nu_e$ from (173) for a rectangular barrier to suggest an alternate form for (189) and the exponential in the Simmons formula:

$$ P_+ \approx N e^{-\frac{8\pi}{3u} \frac{d}{q\hbar\Phi} (2m^*)^{\frac{1}{2}} \left[ \frac{1}{2} E_f \left( 1 + d \sqrt{\frac{2m^*(H-E_f)}{\hbar^2}} \right) \right]^{\frac{3}{2}}} \quad (190) $$

where $V_o \equiv H$ in the calculation of $\lambda'$ in (171) and $\nu_e$ substitutes for $\nu_a$ in (174), with the sine term approximated to 1 to simplify the expression.

To estimate what this formula would need to determine the correct tunneling current, the relative thickness of the oxide, calculated to be a factor of 2.2 less than measured, is assigned to the ratio of the exponent in this theory to that of the Simmons formula:

$$ \left[ \frac{2(H - E_f)}{E_f} \right]^{\frac{3}{2}} \approx \frac{2.2}{u} \quad (191) $$

This suggests that if $\frac{H}{E_f} \approx \frac{1.85}{u}$ in the measurements, this theory would calculate an oxide thickness 2.2 times greater than the Simmons model, ignoring its different determination of the average carrier velocity. There are other considerations that should go into this estimate, such as the details of how the $\lambda_{\pm}$ solutions combine to give an effective $\nu_e$, and to what degree the approximation of the sine term in (174) is valid. However, the dependence on $d$ for (190) has the right trend for a fit to the tapered discrepancy in [34] between the Simmons model and optical measurements for a proper choice of $N$, $E_f$, and $H - E_f$, and $u$.

Further analysis with less approximation may have to be done if more data from experimental work is required to fine tune an accurate tunneling current model. However, the largest takeaway from this study is that there is an alternative to determining oxide thicknesses from standard tunneling formulas which may work better, based on the success in characterizing tunneling current for low bias as well as the analysis done for strong bias which suggests that, aside from normalization and its connection to (176), there isn't as direct a dependence on barrier height for the tunneling probability as conventional analysis has suggested.
$m^*c^2$. Using the same boundary condition $\lambda' = \lambda_{\pm}(\frac{L}{2})$ as before:

$$ \lambda_{\pm}(x) = \pm \frac{4\pi}{3} \left( \frac{E_f d}{q\Phi} + x - \frac{L}{2} \right) + \left( u\lambda_a \mp \frac{4\pi}{3} \frac{E_f d}{q\Phi} \right) \left[ 1 + \frac{q\Phi}{E_f} \frac{x - \frac{L}{2}}{d} \right]^{-\frac{1}{2}} \quad (187) $$

While a full analysis can be performed, the focus here is on a comparison to the exponential probability factors of other approaches such as WKB. Assuming $u\lambda_a \gg \frac{E_f d}{q\Phi}$ or considering high voltages or electric fields, $\frac{E_f}{q\Phi} \ll 1$, to good approximation with the expansion of the square root:

$$ \lambda_{\pm} \approx u\lambda_a \left[ 1 + \frac{q\Phi}{E_f} \frac{x - \frac{L}{2}}{d} \right]^{-\frac{1}{2}} \pm 2\pi \left( x - \frac{L}{2} \right) \quad (188) $$

Ignoring the second term for the moment in considering thin barriers and substituting (188) into (17), the solution takes the following form:

$$ P_+(x=d) \approx N e^{-\frac{8\pi}{3u} \frac{d}{q\hbar\Phi} (2m^*)^{\frac{1}{2}} [E_f + q\Phi]^3} \\ = N e^{-\frac{8\pi}{3u} \frac{d}{q\hbar\Phi} (2m^*)^{\frac{1}{2}} [\frac{1}{2}h\nu_e]^3} \quad (189) $$

where the presence of $\nu_e$ is the result of using (185) at thickness $d$, assuming the $\lambda_{\pm}$ solutions combine to give an effective $\nu_e$.

This formula matches the exponential of the well-known tunneling current analysis of Fowler-Nordheim [33] and Simmons [32] (2.96 replacing 3) for high voltage, if one removes the quantum well, thus setting $u$ to 1 and $H - E_f$ takes the place of $\frac{1}{2}h\nu_e$. What is not shown is how the wavelength and frequency change to bolster the velocity of the carriers as the exponential changes with applied bias. Depending on $V$ and $d$, $\nu_{e-}$ could exponentially increase beyond the node, overshooting saturation to the point that it leads to a premature breakdown of the oxide.

Tunneling current as a function of barrier thickness has been compared in one study [34] to calculations with the Simmons model that predict thicknesses roughly 2.2 times smaller than optical measurements of barriers 20Å thick with an electric field on the order of $15 \times 10^6 \frac{V}{cm}$ (30% from breakdown), and other studies have shown such discrepancies cannot be attributed to defects [35]. Disagreement between the model and experiment tapers slightly to a factor of 2 at 32 Å oxide thickness.

To investigate how this theory's calculation would fair, it first has to be recognized that standard formulas predict an increase in tunneling current with increasing kinetic energy of the carrier. At first, this theory appears to predict the opposite because barrier penetration is more likely for lower frequency components, i.e. longer wavelengths that involve slower evolving phases that permit deeper penetration. However, in this derivation with a

quantum well, what matters is the normalization that is associated with this dependency. The normalization is a function of $u$, which in turn is dependent on the solution to (176). Hence, while the kinetic energy may increase, thus creating a more negative exponent in (189) according to the approximations introduced, $u$ can increase (which is typically seen in finite barrier solutions), thereby reducing the magnitude of the exponent. If it is assumed that the Fermi level is pinned by the bulk, unlike this case study where it was assumed to be affected by adjustment at the barrier, then the current's dependency on $u$ and $\Phi$ or $V$ is clearly dominant.

One can also characterize $\nu_e$ from (173) for a rectangular barrier to suggest an alternate form for (189) and the exponential in the Simmons formula:

$$ P_{+} \approx N e^{-\frac{8\pi}{3u} \frac{d}{q\hbar\Phi} (2m^*)^{\frac{1}{2}} \left[ \frac{1}{2} E_f \left( 1 + d \sqrt{\frac{2m^*(H-E_f)}{\hbar^2}} \right) \right]^{\frac{3}{2}}} \quad (190) $$

where $V_o \equiv H$ in the calculation of $\lambda'$ in (171) and $\nu_e$ substitutes for $\nu_a$ in (174), with the sine term approximated to 1 to simplify the expression.

To estimate what this formula would need to determine the correct tunneling current, the relative thickness of the oxide, calculated to be a factor of 2.2 less than measured, is assigned to the ratio of the exponent in this theory to that of the Simmons formula:

$$ \left[ \frac{2(H - E_f)}{E_f} \right]^{\frac{3}{2}} \approx \frac{2.2}{u} \quad (191) $$

This suggests that if $\frac{H}{E_f} \approx \frac{1.85}{u}$ in the measurements, this theory would calculate an oxide thickness 2.2 times greater than the Simmons model, ignoring its different determination of the average carrier velocity. There are other considerations that should go into this estimate, such as the details of how the $\lambda_{\pm}$ solutions combine to give an effective $\nu_e$, and to what degree the approximation of the sine term in (174) is valid. However, the dependence on $d$ for (190) has the right trend for a fit to the tapered discrepancy in [34] between the Simmons model and optical measurements for a proper choice of $N$, $E_f$, and $H - E_f$, and $u$.

Further analysis with less approximation may have to be done if more data from experimental work is required to fine tune an accurate tunneling current model. However, the largest takeaway from this study is that there is an alternative to determining oxide thicknesses from standard tunneling formulas which may work better, based on the success in characterizing tunneling current for low bias as well as the analysis done for strong bias which suggests that, aside from normalization and its connection to (176), there isn't as direct a dependence on barrier height for the tunneling probability as conventional analysis has suggested.
# X. BOHR MODEL REVISITED

Another derivation of the 1-D equation (8), this time with central force $F = F(r)$, yields the following steady-state equation involving centripetal acceleration at phase equilibrium:

$$F = -\frac{h\nu'}{\lambda_F} = -m_o\frac{v^2}{r} = -m_o\frac{\lambda_e^2\nu_e^2}{r} \quad (192)$$

where $\nu' = \frac{Zk_qq^2}{hr}$ and $\lambda_F = r$ is obtained by equating the Coulomb potential to $-h\nu'$ and the force projection term $\langle\phi'_F|\hat{F}|\phi\rangle$ is equated to unity at resonance. Assuming the resonance condition exists at $\nu_e = \nu'$ whereby quantization can be introduced into the analysis, the following relationship is obtained:

$$\lambda_e^2 = 4\pi^2 r a_o \quad (193)$$

with $a_o = \frac{\hbar^2}{Zk_qq^2m_o}$ as the Bohr radius.

Introducing principal quantum number $n$ by the following ratio, $n^2 = \frac{r}{a_o}$, thus assigning a specific radius to the resonance condition, the wavelength becomes $\lambda_e = 2\pi na_o$, in agreement with the standard wave resonance derivation for the electron's wavelength. This result will be used in obtaining a complete solution for the hydrogen atom in the next section.

# XI. THE HYDROGEN ATOM

Before proceeding with a solution, a few observations are made of the full relativistically covariant analysis of the hydrogen atom as carried out by Dirac, without the inclusion of the interaction of the electron's magnetic moment with the nucleus. The similarity between (74) and (75) suggests $\vec{\alpha}$ and $\beta$, aside from linearising the Hamiltonian, accounts for different components in three dimensions whose evolution is coupled to one another: that of momentum and angular momentum or spin. For example, the quantum operator $\hat{\alpha} \cdot \hat{p}$ represented in spherical coordinates is $-i\hbar\alpha_r \frac{\partial}{\partial r} - \frac{i}{\hbar}\vec{\alpha} \cdot (\frac{\vec{r}}{r} \times \hat{L})$, which expresses the evolution of linear momentum in the radial direction and spin-orbit coupling. The term involving $\beta m_o c^2$, meanwhile, corresponds to the spin's 1-D dynamics characterized earlier and further explored in Appendix A, coupled to the quantum mass in the direction of motion.

Another look is now given to (1) and (58). These expressions refer to one dimension, but considering Dirac's work, the extension to three dimensions suggests a generalization of (1):

$$p = \hbar\chi_{\pm}\tilde{k} = \alpha_i\hat{\chi}_{\pm}\tilde{p}_i \\ = \pm\alpha_i p_i = \pm\alpha_{\parallel}\sqrt{(\hat{\alpha}_i p_i)^2} = \pm\alpha_{\parallel}p_e \quad (194)$$

where an implicit summation convention is used for repeated indices, and explicit use of $e$ notation for the de

Broglie momentum components will be dropped from this point on.

This expression is in Cartesian coordinates, which can be converted to spherical coordinates by a straightforward application of trigonometric functions. While a fully relativistic treatment would suggest the use of amplitudes for the equations of motion, Newton's modified law, or in general terms, Lagrangian dynamics [36], is now applied to the hydrogen atom, generating the following equation using de Broglie components:

$$\frac{d}{dt}(\hat{\alpha}_i m \gamma) \vec{v}_i + \hat{\alpha}_i m \gamma \vec{a}_i = -\frac{Zk_q q^2}{r^2} \langle \phi'_F | \hat{F} | \phi \rangle \hat{r} \quad (195)$$

which can be derived in the same manner as done for 1-D problems by first substituting de Broglie's relation for the momentum (1) with $\vec{p} = \hat{\chi}_{\pm} \frac{\hbar \vec{v}}{\lambda_e \vec{v}} = \hat{\alpha}_i m \gamma \vec{v}_i$.

That recreates (6) using $v = \lambda_e \nu_e$, so it is just as valid to directly substitute quantum mass $m$ for $m_o$ into an otherwise semi-classical expression with corrections due to the dynamics of spin in three dimensions. As an aside, the reduced mass $\mu$ of the electron with the nucleus can be viewed as implicit with $m$. Previous works [37], [38] derived Schrödinger's equation from Newton's 2nd law using an additional force term representing diffusion-type processes, but this approach isn't applied here.

An expansion of (195) in spherical harmonics would introduce coupling between different $\hat{\alpha}_i$ coefficients and components of angular momentum, as expected. However, a more detailed modeling of spin is required to carry this particular analysis further, which is not presented in this work. Instead, an analysis closer in alignment with Schrödinger's approach will be pursued, with the assumption that $\hat{\alpha}_i \approx 1$.

Unlike the tunneling current problems, the evaluation of $\langle\phi'_F|\hat{F}|\phi\rangle$ does not need to be statistically averaged. Assuming it is a function of position only and that $\hat{F} = \hat{\chi}_{\pm}$, it is brought to the LHS of (195). The notation from this point on will make the following assignment:

$$m \leftrightarrow m_{\phi} \quad (196)$$

in accordance with the discussion surrounding (118).

Expanding out (195) in spherical coordinates with azimuthal angle $\phi$ according to the well-known expansion of Newton's 2nd law, the following set of equations are obtained:

$$\left(\frac{d}{dt}m\gamma\right)\dot{r} + m\gamma\left(\ddot{r} - r\dot{\theta}^2 - r\dot{\phi}^2\sin^2\theta\right) = -\frac{Zk_q q^2}{r^2} \quad (197)$$

$$\left(\frac{d}{dt}m\gamma\right)\dot{r} + m\gamma\left(r\ddot{\theta} + 2\dot{r}\dot{\theta} - r\dot{\phi}^2\sin\theta\cos\theta\right) = 0 \quad (198)$$

$$\left(\frac{d}{dt}m\gamma\right)rsin\theta\dot{\phi} + \frac{m\gamma}{rsin\theta}\frac{d}{dt}\left(r^2sin^2\theta\dot{\phi}\right) = 0 \quad (199)$$

Multiplying (199) by $rsin\theta$ identifies the first Noether's [39] conserved parameter that would invite quantization.

$$K_{\theta} \equiv m\gamma r^2 \sin^2\theta \dot{\phi} = -L_{\theta} \sin\theta \equiv m_{\theta} \cdot \hbar \quad (200)$$
## X. BOHR MODEL REVISITED

Another derivation of the 1-D equation (8), this time with central force $F = F(r)$, yields the following steady-state equation involving centripetal acceleration at phase equilibrium:

$$F = -\frac{h\nu'}{\lambda_F} = -m_o \frac{v^2}{r} = -m_o \frac{\lambda_e^2 \nu_e^2}{r} \quad (192)$$

where $\nu' = \frac{Zk_q q^2}{hr}$ and $\lambda_F = r$ is obtained by equating the Coulomb potential to $-h\nu'$ and the force projection term $\langle\phi'_F|\hat{F}|\phi\rangle$ is equated to unity at resonance. Assuming the resonance condition exists at $\nu_e = \nu'$ whereby quantization can be introduced into the analysis, the following relationship is obtained:

$$\lambda_e^2 = 4\pi^2 r a_o \quad (193)$$

with $a_o = \frac{\hbar^2}{Zk_q q^2 m_o}$ as the Bohr radius.

Introducing principal quantum number $n$ by the following ratio, $n^2 = \frac{r}{a_o}$, thus assigning a specific radius to the resonance condition, the wavelength becomes $\lambda_e = 2\pi na_o$, in agreement with the standard wave resonance derivation for the electron's wavelength. This result will be used in obtaining a complete solution for the hydrogen atom in the next section.

## XI. THE HYDROGEN ATOM

Before proceeding with a solution, a few observations are made of the full relativistically covariant analysis of the hydrogen atom as carried out by Dirac, without the inclusion of the interaction of the electron's magnetic moment with the nucleus. The similarity between (74) and (75) suggests $\vec{\alpha}$ and $\beta$, aside from linearising the Hamiltonian, accounts for different components in three dimensions whose evolution is coupled to one another: that of momentum and angular momentum or spin. For example, the quantum operator $\hat{\alpha} \cdot \hat{p}$ represented in spherical coordinates is $-i\hbar\alpha_r \frac{\partial}{\partial r} - \frac{i}{\hbar}\vec{\alpha} \cdot (\frac{\vec{r}}{r} \times \hat{L})$, which expresses the evolution of linear momentum in the radial direction and spin-orbit coupling. The term involving $\beta m_o c^2$, meanwhile, corresponds to the spin's 1-D dynamics characterized earlier and further explored in Appendix A, coupled to the quantum mass in the direction of motion.

Another look is now given to (1) and (58). These expressions refer to one dimension, but considering Dirac's work, the extension to three dimensions suggests a generalization of (1):

$$p = \hbar\chi_{\pm}\tilde{k} = \alpha_i\hat{\chi}_{\pm}\tilde{p}_i = \pm\alpha_i p_i = \pm\alpha_{\parallel}\sqrt{(\hat{\alpha}_i p_i)^2} = \pm\alpha_{\parallel}p_e \quad (194)$$

where an implicit summation convention is used for repeated indices, and explicit use of $e$ notation for the de

Broglie momentum components will be dropped from this point on.

This expression is in Cartesian coordinates, which can be converted to spherical coordinates by a straightforward application of trigonometric functions. While a fully relativistic treatment would suggest the use of amplitudes for the equations of motion, Newton's modified law, or in general terms, Lagrangian dynamics [36], is now applied to the hydrogen atom, generating the following equation using de Broglie components:

$$\frac{d}{dt}(\hat{\alpha}_i m \gamma) \vec{v}_i + \hat{\alpha}_i m \gamma \vec{a}_i = - \frac{Z k_q q^2}{r^2} \langle \phi'_F | \hat{F} | \phi \rangle \hat{r} \quad (195)$$

which can be derived in the same manner as done for 1-D problems by first substituting de Broglie's relation for the momentum (1) with $\vec{p} = \hat{\chi}_{\pm} \frac{\hbar \vec{v}}{\lambda_e \vec{v}} = \hat{\alpha}_i m \gamma \vec{v}_i$.

That recreates (6) using $v = \lambda_e \nu_e$, so it is just as valid to directly substitute quantum mass $m$ for $m_o$ into an otherwise semi-classical expression with corrections due to the dynamics of spin in three dimensions. As an aside, the reduced mass $\mu$ of the electron with the nucleus can be viewed as implicit with $m$. Previous works [37], [38] derived Schrödinger's equation from Newton's 2nd law using an additional force term representing diffusion-type processes, but this approach isn't applied here.

An expansion of (195) in spherical harmonics would introduce coupling between different $\hat{\alpha}_i$ coefficients and components of angular momentum, as expected. However, a more detailed modeling of spin is required to carry this particular analysis further, which is not presented in this work. Instead, an analysis closer in alignment with Schrödinger's approach will be pursued, with the assumption that $\hat{\alpha}_i \approx 1$.

Unlike the tunneling current problems, the evaluation of $\langle\phi'_F|\hat{F}|\phi\rangle$ does not need to be statistically averaged. Assuming it is a function of position only and that $\hat{F} = \hat{\chi}_{\pm}$, it is brought to the LHS of (195). The notation from this point on will make the following assignment:

$$m \leftrightarrow m_{\phi} \quad (196)$$

in accordance with the discussion surrounding (118).

Expanding out (195) in spherical coordinates with azimuthal angle $\phi$ according to the well-known expansion of Newton's 2nd law, the following set of equations are obtained:

$$\left(\frac{d}{dt}m\gamma\right)\dot{r} + m\gamma\left(\ddot{r} - r\dot{\theta}^2 - r\dot{\phi}^2\sin^2\theta\right) = -\frac{Zk_q q^2}{r^2} \quad (197)$$

$$\left(\frac{d}{dt}m\gamma\right)\dot{r} + m\gamma\left(r\ddot{\theta} + 2\dot{r}\dot{\theta} - r\dot{\phi}^2\sin\theta\cos\theta\right) = 0 \quad (198)$$

$$\left(\frac{d}{dt}m\gamma\right)rsin\theta\dot{\phi} + \frac{m\gamma}{rsin\theta}\frac{d}{dt}(r^2sin^2\theta\dot{\phi}) = 0 \quad (199)$$

Multiplying (199) by $rsin\theta$ identifies the first Noether's [39] conserved parameter that would invite quantization.

$$K_{\theta} \equiv m\gamma r^2 \sin^2\theta \dot{\phi} = -L_{\theta} \sin\theta \equiv m_{\theta} \cdot \hbar \quad (200)$$
where $L_\theta$ is synonymous with the notation $L_z$ commonly used in the presentation of Schrödinger's solution. Given the absence of torque for the Coulomb force, the angular momentum $\vec{r} \times \vec{p}$ and its square is also conserved and quantized:

$$L^2 = |\vec{r} \times \vec{p}|^2 = \left| m\gamma r^2 \dot{\theta}\dot{\phi} - m\gamma r^2 \sin\theta \dot{\phi}\dot{\theta} \right|^2 \\ = L_{\phi}^2 + L_{\theta}^2 = l(l+1)\hbar^2 \quad (201)$$

with quantum number $l$ introduced to match Schrödinger's result and $L_\phi \equiv m\gamma r^2 \dot{\theta}$.

By using (6) and forming the following relations:

$$\begin{aligned} v^2 &= \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \dot{\phi}^2 \sin^2 \theta \\ r\ddot{r} &= \frac{d}{dt}(r\dot{r}) - \dot{r}^2 \end{aligned} \quad (202)$$

a regrouping of terms in (197) can be performed after first multiplying by $r$:

$$r\dot{r}\frac{d}{dt}(m\gamma)+m\gamma\left(\frac{d}{dt}(r\dot{r})-\dot{r}^2-r^2\dot{\theta}^2-\dot{r}\dot{\phi}^2\sin^2\theta\right)=-h\nu' \quad (203)$$

where $\nu' = \frac{Zk_qq^2}{hr}$ is identified along with $\lambda_F = r$.

Then from (6), $h\nu_e = m\gamma v^2$, yielding the following steady-state equation (see Appendix B for its relationship to the eigenvalue problem):

$$\frac{d}{dt}(m\gamma r\dot{r}) = \nabla(m\gamma r\dot{r}) \cdot \vec{v} = h(\nu_e - \nu') \quad (204)$$

As an aside:

$$L_r \equiv m\gamma r\dot{r} \quad (205)$$

represents the component absent from $\vec{r} \times \vec{p}$ and $\nu_e = \nu'$ identifies a resonance condition with $L_r$ stationary at $r = n^2 a_o$ (examined below). The quantity $\lambda_F \nu'$, which equals $3.48 \times 10^5 \text{m/s}$ for $Z = 1$, might suggest a slow-down of electromagnetic propagation for small $Z$ and $r$, facilitating the electron's absorption of its own radiation in orbit as represented by the lowest-order Feynman diagram characterizing the Lamb shift. Further analysis would be needed to see how quantum electrodynamics (QED) could be reconciled with the work in this paper.

Since $\nu_e$ responds only to the Coulomb force $F(r)$ in steady-state, $\nu_e = \nu(r)$; from (204) and (208), $\lambda_e^2 = \lambda_e^2(r)$, with the same dependence for $m\gamma$. This permits the following generalized Bohr relation to be used to introduce principal quantum number $n$ into the analysis, supported by further derivation below:

$$|\vec{v}| = \frac{[L_{\theta}^2 + L_z^2 + L_r^2]^{\frac{1}{2}}}{m\gamma r} \stackrel{!}{=} \frac{n\hbar}{m\gamma r} \quad \left\{ m\gamma \right\}_{r=n^2 a_o} = m_o \quad (206)$$

with Bohr's original quantization constraint holding at $r = n^2 a_o$ [40]. Upon substitution of (6) into (206), $\lambda_e(r) = 2\pi n a_o$, in agreement with (193).

To solve (204), the $r$ dependence of $\dot{r}$ is obtained by re-expressing (202) in terms of (6) and (201). This is done by first recognizing the following relationship:

$$p_e^2 r^2 - L_r^2 = L_{\phi}^2 + L_{\theta}^2 = l(l+1)\hbar^2 \quad (207)$$

This expression has the right behavior if $p_e$ is orthogonal to $\vec{r}$, setting $L_r = 0$ and if $p_e$ is parallel to $\vec{r}$, setting $L_\phi^2 + L_\theta^2 = 0$. The following is then readily shown:

$$\dot{r} = \pm \frac{1}{m\gamma r} \left[ \frac{h^2}{\lambda_e^2} r^2 - l(l+1)\hbar^2 \right]^{\frac{1}{2}} \quad (208)$$

With cancellations from applying the chain rule, and multiplying through by 2 times the $\frac{1}{2}$ part of (208) after taking its derivative to isolate the $\theta$, $\phi$ dependence, (204) becomes:

$$\begin{aligned} & \left[ \frac{h^2}{\lambda_e^2} r^2 - l(l+1)\hbar^2 \right]^{\frac{1}{2}} r \frac{\partial}{\partial r} \frac{h}{\lambda_e^2} \pm \left[ L_{\phi} \frac{\partial}{\partial \theta} + \frac{K_{\theta}}{\sin^2 \theta} \frac{\partial}{\partial \phi} \right] \frac{h}{\lambda_e^2} \\ & = -2 \left[ \frac{h^2}{\lambda_e^2} r^2 - l(l+1)\hbar^2 \right]^{\frac{1}{2}} \frac{\nu'}{\nu} \frac{h}{\lambda_e^2} \end{aligned} \quad (209)$$

Defining a variable Bohr radius $\tilde{a}_o$:

$$\tilde{a}_o \equiv \frac{\hbar^2}{m\gamma Z k_q q^2} \qquad \xi \equiv \frac{r}{n\tilde{a}_o} \quad (210)$$

and position variable $\xi$ will now be used to generate an equation for $P(\xi)$. As the electron's mesoscopic characterization by radius $r$ extends over a sphere, $\alpha_\parallel$ coefficients are associated with the $\theta$ and $\phi$ variables, presented further down, and $P(\xi)$ is obtained from (14) by first integrating over the solid angle, initially expressed with position variable $r$:

$$\begin{aligned} \frac{dP(\xi)}{dr} &= -2k_e \int P(r, \theta, \phi) r^2 d\Omega \\ &= -2k_e (4\pi r^2 P(r)) \equiv -2k_e P(\xi) \end{aligned} \quad (211)$$

This transitions the domain of problems with cylindrical symmetry to those with probabilities residing in spherical shells, specified by $4\pi r^2 P(r) dr$. Also, the $\pm$ introduced by the $\hat{\chi}_\pm$ operator is restricted here to the negative sign for an attractive force. Recognizing $k_e = \frac{2\pi\nu_e}{\nu}$ and that $\nu$ is given by (206), along with using the chain rule $\frac{d}{dr} = \frac{d}{d\xi}$ where $\xi$ is defined by (210), (211) takes the following basic form:

$$\frac{dP}{d\xi} = -\frac{\nu_e(\xi)}{\nu'} P(\xi) \quad (212)$$

where a factor of 2 is incorporated into the definition of $\nu_e$ (see below).

Introducing separation of variables into (209), with $\lambda = \lambda(\xi)\Theta^{-1}(\theta)\Phi^{-1}(\phi)$, while substituting in $\xi$ and (6), an ordinary differential equation is obtained:

$$\xi \frac{d(m\gamma\nu_e)}{d\xi} = -2 \frac{\nu'}{\nu_e} m\gamma\nu_e \quad (213)$$
where $L_\theta$ is synonymous with the notation $L_z$ commonly used in the presentation of Schrödinger's solution. Given the absence of torque for the Coulomb force, the angular momentum $\vec{r} \times \vec{p}$ and its square is also conserved and quantized:

$$L^2 = |\vec{r} \times \vec{p}|^2 = \left| m\gamma r^2 \dot{\theta}\dot{\phi} - m\gamma r^2 \sin\theta \dot{\phi}\dot{\theta} \right|^2 \\ = L_{\phi}^2 + L_{\theta}^2 = l(l+1)\hbar^2 \qquad (201)$$

with quantum number $l$ introduced to match Schrödinger's result and $L_\phi \equiv m\gamma r^2 \dot{\theta}$.

By using (6) and forming the following relations:

$$\begin{aligned} v^2 &= \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \dot{\phi}^2 \sin^2 \theta \\ r\ddot{r} &= \frac{d}{dt}(r\dot{r}) - \dot{r}^2 \end{aligned} \qquad (202)$$

a regrouping of terms in (197) can be performed after first multiplying by $r$:

$$r\dot{r}\frac{d}{dt}(m\gamma)+m\gamma\left(\frac{d}{dt}(r\dot{r})-\dot{r}^2-r^2\dot{\theta}^2-\dot{r}\dot{\phi}^2\sin^2\theta\right)=-h\nu' \qquad (203)$$

where $\nu' = \frac{Zk_qq^2}{hr}$ is identified along with $\lambda_F = r$.

Then from (6), $h\nu_e = m\gamma v^2$, yielding the following steady-state equation (see Appendix B for its relationship to the eigenvalue problem):

$$\frac{d}{dt}(m\gamma r\dot{r}) = \nabla(m\gamma r\dot{r}) \cdot \vec{v} = h(\nu_e - \nu') \qquad (204)$$

As an aside:

$$L_r \equiv m\gamma r\dot{r} \qquad (205)$$

represents the component absent from $\vec{r} \times \vec{p}$ and $\nu_e = \nu'$ identifies a resonance condition with $L_r$ stationary at $r = n^2 a_o$ (examined below). The quantity $\lambda_F \nu'$, which equals $3.48 \times 10^5 \text{m/s}$ for $Z = 1$, might suggest a slow-down of electromagnetic propagation for small $Z$ and $r$, facilitating the electron's absorption of its own radiation in orbit as represented by the lowest-order Feynman diagram characterizing the Lamb shift. Further analysis would be needed to see how quantum electrodynamics (QED) could be reconciled with the work in this paper.

Since $\nu_e$ responds only to the Coulomb force $F(r)$ in steady-state, $\nu_e = \nu(r)$; from (204) and (208), $\lambda_e^2 = \lambda_e^2(r)$, with the same dependence for $m\gamma$. This permits the following generalized Bohr relation to be used to introduce principal quantum number $n$ into the analysis, supported by further derivation below:

$$|\vec{v}| = \frac{[L_{\theta}^2 + L_z^2 + L_r^2]^{\frac{1}{2}}}{m\gamma r} \stackrel{!}{=} \frac{n\hbar}{m\gamma r} \quad \left\{ m\gamma \right\}_{r=n^2 a_o} = m_o \qquad (206)$$

with Bohr's original quantization constraint holding at $r = n^2 a_o$ [40]. Upon substitution of (6) into (206), $\lambda_e(r) = 2\pi n a_o$, in agreement with (193).

To solve (204), the $r$ dependence of $\dot{r}$ is obtained by re-expressing (202) in terms of (6) and (201). This is done by first recognizing the following relationship:

$$p_e^2 r^2 - L_r^2 = L_{\phi}^2 + L_{\theta}^2 = l(l+1)\hbar^2 \qquad (207)$$

This expression has the right behavior if $p_e$ is orthogonal to $\vec{r}$, setting $L_r = 0$ and if $p_e$ is parallel to $\vec{r}$, setting $L_\phi^2 + L_\theta^2 = 0$. The following is then readily shown:

$$\dot{r} = \pm \frac{1}{m\gamma r} \left[ \frac{h^2}{\lambda_e^2} r^2 - l(l+1)\hbar^2 \right]^{\frac{1}{2}} \qquad (208)$$

With cancellations from applying the chain rule, and multiplying through by 2 times the $\frac{1}{2}$ part of (208) after taking its derivative to isolate the $\theta$, $\phi$ dependence, (204) becomes:

$$\begin{aligned} & \left[ \frac{h^2}{\lambda_e^2} r^2 - l(l+1)\hbar^2 \right]^{\frac{1}{2}} r \frac{\partial}{\partial r} \frac{h}{\lambda_e^2} \pm \left[ L_{\phi} \frac{\partial}{\partial \theta} + \frac{K_{\theta}}{\sin^2 \theta} \frac{\partial}{\partial \phi} \right] \frac{h}{\lambda_e^2} \\ & = -2 \left[ \frac{h^2}{\lambda_e^2} r^2 - l(l+1)\hbar^2 \right]^{\frac{1}{2}} \frac{\nu'}{\nu} \frac{h}{\lambda_e^2} \end{aligned} \qquad (209)$$

Defining a variable Bohr radius $\tilde{a}_o$:

$$\tilde{a}_o \equiv \frac{\hbar^2}{m\gamma Z k_q q^2} \qquad \xi \equiv \frac{r}{n\tilde{a}_o} \qquad (210)$$

and position variable $\xi$ will now be used to generate an equation for $P(\xi)$. As the electron's mesoscopic characterization by radius $r$ extends over a sphere, $\alpha_\parallel$ coefficients are associated with the $\theta$ and $\phi$ variables, presented further down, and $P(\xi)$ is obtained from (14) by first integrating over the solid angle, initially expressed with position variable $r$:

$$\begin{aligned} \frac{dP(\xi)}{dr} &= -2k_e \int P(r, \theta, \phi) r^2 d\Omega \\ &= -2k_e (4\pi r^2 P(r)) \equiv -2k_e P(\xi) \end{aligned} \qquad (211)$$

This transitions the domain of problems with cylindrical symmetry to those with probabilities residing in spherical shells, specified by $4\pi r^2 P(r)dr$. Also, the $\pm$ introduced by the $\hat{\chi}_\pm$ operator is restricted here to the negative sign for an attractive force. Recognizing $k_e = \frac{2\pi\nu_e}{\nu}$ and that $\nu$ is given by (206), along with using the chain rule $\frac{d}{dr} = \frac{d}{d\xi}$ where $\xi$ is defined by (210), (211) takes the following basic form:

$$\frac{dP}{d\xi} = - \frac{\nu_e(\xi)}{\nu'} P(\xi) \qquad (212)$$

where a factor of 2 is incorporated into the definition of $\nu_e$ (see below).

Introducing separation of variables into (209), with $\lambda = \lambda(\xi)\Theta^{-1}(\theta)\Phi^{-1}(\phi)$, while substituting in $\xi$ and (6), an ordinary differential equation is obtained:

$$\xi \frac{d(m\gamma\nu_e)}{d\xi} = -2 \frac{\nu'}{\nu_e} m\gamma\nu_e \qquad (213)$$
with the following solution:

$$m\gamma\nu_e = \frac{h}{\lambda_e^2} = Ne^{-2\int \frac{\nu_e'}{\nu_e} \frac{d\xi}{\xi}} \quad (214)$$

Given $\lambda_e$ can be complex with $\lambda_e^2 = \lambda_e^*\lambda_e$ in $\mathbb{R}$, any normalized complex functions for $\Theta(\theta)\Phi(\phi)$ would satisfy (209) since, in effect, $\lambda_e^2 = \lambda_e^2(\xi)$ and:

$$\left[ L_{\phi} \frac{\partial}{\partial \theta} + \frac{K_{\theta}}{\sin^2 \theta} \frac{\partial}{\partial \phi} \right] \Theta^2(\theta) \Phi^2(\phi) = 0 \quad (215)$$

Hence, if Schrödinger's representation of $\psi$'s angular dependence is correct, the expressions for $k_\theta$ and $k_\phi$ in $\mathbb{R}$, corresponding to $\Theta(\theta)$ and $\Phi(\phi)$ expressed in $\mathbb{C}$, would be determined from the probabilities associated with spherical harmonics by solving the following equations consistent with (16) and (17) for the respective variables:

$$\frac{dP_l^{m_l}(\theta)}{d\theta} = \pm 2\alpha_{ml}^l k_\theta(\theta) P_l^{m_l}(\theta) \quad (216)$$

$$\frac{dP_l^{m_l}(\phi)}{d\phi} = \pm 2\alpha_{ml}^l k_\phi(\phi) P_l^{m_l}(\phi) \quad (217)$$

with the quantization of $K_\theta$ and the definition of $m_\theta$ ignored.

However, given a preference to use $K_\theta$ to quantize due to its conservation classically over $L_z$ defined in Schrödinger's approach, it is tempting to solve (215) with a separation of variables that generates the following two equations, using the result of (200):

$$K_{\theta} \frac{\partial \Phi(\phi)}{\partial \phi} = \pm inm_{\theta} \hbar \Phi(\phi) \quad (218)$$

$$L_{\phi} \sin^2 \theta \frac{\partial \Theta(\theta)}{\partial \theta} = \mp inm_{\theta} \hbar \Theta(\theta) \quad (219)$$

where the alignment of $K_\theta$ with the derivative in $\phi$ is due to its dependence on $\phi$ and likewise for $L_\phi$ for its dependence on $\dot{\theta}$. While $K_\theta$ is a quantized quantity, $L_\phi$ is an operator whose determination would be tied to the considerations of spin and generalized angular momentum $\mathbf{J}$.

Solving for $\Phi(\phi)$, one obtains its representation in $\mathbb{C}$:

$$\Phi(\phi) = A_{\phi} e^{\pm in\phi} \quad (220)$$

which imposes the constraint on $n$ that it be an integer in order for $\Phi(0) = \Phi(2\pi)$. Recognizing this solution as the equivalent of (17) and using (16) to determine the probability, one obtains:

$$P_n(\phi) = N \cos^2(n\phi) \quad (221)$$

which in turn generates the corresponding azimuthal component of $k = k_r k_\phi k_\theta$ in $\mathbb{R}$ matching up with $k_r = 2\pi n a_o$ at resonance. This gives a solution to (217) where $m_l$ and $\alpha_{m_l}$ are replaced by $n$:

$$k_{\phi}(\phi) = \tan(n\phi) \quad (222)$$

Here it is seen that the Bohr criterion for quantizing the resonance condition, i.e. the introduction of $n$, is attributable to $k_\phi$ acting as the phase component and $k_r$ acting as the amplitude, which is identical to a free-particle solution with the same method of derivation, only with a periodicity determined by the quantization introduced by azimuthal symmetry. The quantum number $n$ determines the energy content due to its presence in $k_r$ while in turn, $k_\phi$ mirrors the wavelength's scale of variance in the resonant orbit, with the values of $L_\theta$ and $L_\phi$ largely determined by the other two quantum numbers $l$ and $m_\theta$. Two key problems attributed to classical descriptions of the hydrogen atom will now be discussed. The first has to do with why orbits do not remain planar and the second has to do with why the electron does not radiate energy.

The orbits are non-planar due to the coupling between angular momentum and spin along with the spin's coupling to a virtual force. As discussed for a constant applied force, a virtual force exists in tandem with an applied force that maintains a particle's wavelike dynamics as long as the exchange energy with the background, $\nu_x$, does not exceed the background's capacity, thus preserving phase coherence. Given the spherical symmetry of the hydrogen atom, quantization stabilizes the energy such that exchanges with the background are bounded, and hence phase coherence is preserved. As such, the spin's dynamics introduce virtual forces that do not point along the line connecting the electron with the nucleus. It is also worth noting that $|K_\theta|$ can be preserved with a varying $\theta$ component and likewise, $|L_\phi|$ can be preserved with a varying $\phi$ component.

As to why the electron does not radiate energy, QED has been used to explain this but with the understanding that the electromagnetic field surrounding the atom and the atom itself form a quantum system. Given the formulation of the free particle solution in this work, it is suggested that this QED argument can be narrowed to the phase bubble that comprises the wave energy of the particle. Just as the varying velocity of a "free", charged particle with phase coherence preserved does not cause the particle to radiate, so does the particle not radiate if its phase coherence is preserved as it travels around the nucleus. When the phase bubble is unstable, i.e. when it is either absorbing or emitting energy in a fashion that goes outside the mitigating influence of the background, then phase coherence is destroyed in that moment and the particle together with the background E&M field behave classically. Though this sounds like wavefunction collapse, the difference is that the particle's trajectory remains continuous.

Further investigation is needed into the use of generalized angular momentum $\mathbf{J}$ to determine the appropriate way to diverge from Schrödinger's original analysis. Only precise measurements [12] of $l=0$ states are available for comparison, and for this reason further analysis of the angular dependence is not pursued in this work. As to whether or not a rarification of density associated with
with the following solution:

$$m\gamma\nu_e = \frac{h}{\lambda_e^2} = Ne^{-2\int \frac{\nu_e'}{\nu_e} \frac{d\xi}{\xi}} \quad (214)$$

Given $\lambda_e$ can be complex with $\lambda_e^2 = \lambda_e^*\lambda_e$ in $\mathbb{R}$, any normalized complex functions for $\Theta(\theta)\Phi(\phi)$ would satisfy (209) since, in effect, $\lambda_e^2 = \lambda_e^2(\xi)$ and:

$$\left[ L_{\phi} \frac{\partial}{\partial \theta} + \frac{K_{\theta}}{\sin^2 \theta} \frac{\partial}{\partial \phi} \right] \Theta^2(\theta) \Phi^2(\phi) = 0 \quad (215)$$

Hence, if Schrödinger's representation of $\psi$'s angular dependence is correct, the expressions for $k_\theta$ and $k_\phi$ in $\mathbb{R}$, corresponding to $\Theta(\theta)$ and $\Phi(\phi)$ expressed in $\mathbb{C}$, would be determined from the probabilities associated with spherical harmonics by solving the following equations consistent with (16) and (17) for the respective variables:

$$\frac{dP_l^{m_l}(\theta)}{d\theta} = \pm 2\alpha_{ml}^l k_\theta(\theta) P_l^{m_l}(\theta) \quad (216)$$

$$\frac{dP_l^{m_l}(\phi)}{d\phi} = \pm 2\alpha_{ml}^l k_\phi(\phi) P_l^{m_l}(\phi) \quad (217)$$

with the quantization of $K_\theta$ and the definition of $m_\theta$ ignored.

However, given a preference to use $K_\theta$ to quantize due to its conservation classically over $L_z$ defined in Schrödinger's approach, it is tempting to solve (215) with a separation of variables that generates the following two equations, using the result of (200):

$$K_{\theta} \frac{\partial \Phi(\phi)}{\partial \phi} = \pm inm_{\theta} \hbar \Phi(\phi) \quad (218)$$

$$L_{\phi} \sin^2 \theta \frac{\partial \Theta(\theta)}{\partial \theta} = \mp inm_{\theta} \hbar \Theta(\theta) \quad (219)$$

where the alignment of $K_\theta$ with the derivative in $\phi$ is due to its dependence on $\phi$ and likewise for $L_\phi$ for its dependence on $\dot{\theta}$. While $K_\theta$ is a quantized quantity, $L_\phi$ is an operator whose determination would be tied to the considerations of spin and generalized angular momentum $\mathbf{J}$.

Solving for $\Phi(\phi)$, one obtains its representation in $\mathbb{C}$:

$$\Phi(\phi) = A_{\phi} e^{\pm in\phi} \quad (220)$$

which imposes the constraint on $n$ that it be an integer in order for $\Phi(0) = \Phi(2\pi)$. Recognizing this solution as the equivalent of (17) and using (16) to determine the probability, one obtains:

$$P_n(\phi) = N \cos^2(n\phi) \quad (221)$$

which in turn generates the corresponding azimuthal component of $k = k_r k_\phi k_\theta$ in $\mathbb{R}$ matching up with $k_r = 2\pi n a_o$ at resonance. This gives a solution to (217) where $m_l$ and $\alpha_{m_l}$ are replaced by $n$:

$$k_{\phi}(\phi) = \tan(n\phi) \quad (222)$$

Here it is seen that the Bohr criterion for quantizing the resonance condition, i.e. the introduction of $n$, is attributable to $k_\phi$ acting as the phase component and $k_r$ acting as the amplitude, which is identical to a free-particle solution with the same method of derivation, only with a periodicity determined by the quantization introduced by azimuthal symmetry. The quantum number $n$ determines the energy content due to its presence in $k_r$ while in turn, $k_\phi$ mirrors the wavelength's scale of variance in the resonant orbit, with the values of $L_\theta$ and $L_\phi$ largely determined by the other two quantum numbers $l$ and $m_\theta$. Two key problems attributed to classical descriptions of the hydrogen atom will now be discussed. The first has to do with why orbits do not remain planar and the second has to do with why the electron does not radiate energy.

The orbits are non-planar due to the coupling between angular momentum and spin along with the spin's coupling to a virtual force. As discussed for a constant applied force, a virtual force exists in tandem with an applied force that maintains a particle's wavelike dynamics as long as the exchange energy with the background, $\nu_x$, does not exceed the background's capacity, thus preserving phase coherence. Given the spherical symmetry of the hydrogen atom, quantization stabilizes the energy such that exchanges with the background are bounded, and hence phase coherence is preserved. As such, the spin's dynamics introduce virtual forces that do not point along the line connecting the electron with the nucleus. It is also worth noting that $|K_\theta|$ can be preserved with a varying $\theta$ component and likewise, $|L_\phi|$ can be preserved with a varying $\phi$ component.

As to why the electron does not radiate energy, QED has been used to explain this but with the understanding that the electromagnetic field surrounding the atom and the atom itself form a quantum system. Given the formulation of the free particle solution in this work, it is suggested that this QED argument can be narrowed to the phase bubble that comprises the wave energy of the particle. Just as the varying velocity of a "free", charged particle with phase coherence preserved does not cause the particle to radiate, so does the particle not radiate if its phase coherence is preserved as it travels around the nucleus. When the phase bubble is unstable, i.e. when it is either absorbing or emitting energy in a fashion that goes outside the mitigating influence of the background, then phase coherence is destroyed in that moment and the particle together with the background E&M field behave classically. Though this sounds like wavefunction collapse, the difference is that the particle's trajectory remains continuous.

Further investigation is needed into the use of generalized angular momentum $\mathbf{J}$ to determine the appropriate way to diverge from Schrödinger's original analysis. Only precise measurements [12] of $l = 0$ states are available for comparison, and for this reason further analysis of the angular dependence is not pursued in this work. As to whether or not a rarification of density associated with
the $cos^2(n\phi)$ probability calculation was observed in [12], it should be noted that observations were the result of a collection of photoelectrons from many atoms, hence the orbits represent a statistical average over variances in phase offsets. Also, virtual forces scramble the $\theta$, $\phi$ dependence of the angular momentum/probability over time, and an applied electric field will have an impact.

To solve for the two variables in (214), an observation from Schrödinger theory is used: The solution for $P(\xi)$ inside the Coulomb potential well for a given energy, with similarity to the square well solution in terms of the generation of nodes, will have unique degenerate solutions with (0,1,...,n-1) zeros at or away from the origin denoted by quantum number $l$, and a 2nd order zero at the origin attributed to $r^2$ from integration over the solid angle of Schrödinger's probability: $P(\xi) = r^2 \int \psi^* \psi d\Omega$. Given the form of (212), this means that the zeros of $P(\xi)$ will appear as poles of $\nu_e(\xi)$.

Thus, considering what is known so far, including also the factor of 2 mentioned previously, $\nu_e$ takes the following general form with constants $\delta$, $c_j$, and original zero $s_j$ of $P(r)$:

$$ \nu_e(\xi) = 2\nu' \left( \delta - \frac{(l+1)}{\xi} - \sum_{j=1}^{n-(l+1)} \mathbf{Z}_j \right) \quad (223) $$

where $2^{l+1}/\xi$ is due to the zero of $P(\xi)$ at the origin with

$$ \mathbf{Z}_j = \frac{1}{c_j \xi - \frac{a_o}{a_o} s_j} \quad (224) $$

representing the poles of $\nu$ away from the origin and $a_o/a_o$ is introduced when replacing $r/a_o$ by $\xi$.

Specific cases are now examined, starting with $n, l = n-1$ where $\nu = 2\nu'(\delta - \frac{n}{\xi})$. Hypothetically, if $\delta = 1$, $\tilde{a}_o = a_o$, and $\xi = \frac{r}{na_o} = n$ at the resonance radii of Bohr's analysis, (212) generates the probability:

$$ P_{n,n-1}(\xi) = N^2 \xi^{2n} e^{-2\xi} \quad (225) $$

which in absorbing a factor of $n$ into the normalization agrees with the square of solutions to Schrödinger's radial equation multiplied by $r^2$. The next case is $n, l = n-2$ in which (223) has form $\nu_e = 2\nu'(1 - \frac{n-1}{\xi} - \frac{1}{x_1\xi - s_1})$ using the same hypothesis as before. Setting $\nu_e = 0$ at $\xi = n$ to be in line with the $n, l = n-1$ case, $c_1 = n$ and $s_1 = n^2 - n$. Solving (212):

$$ P_{n,n-2}(\xi) = N^2 \xi^{2(n-1)} [n\xi - (n^2 - n)]^2 e^{-2\xi} \quad (226) $$

again, matching the solutions to Schrödinger's radial equation multiplied by $r^2$ with a factor of $n$ absorbed by the normalization. This suggests, in principle, boundary conditions can be chosen to enforce agreement with Schrödinger's formulation, however there would be contradictions.

First, as $\xi$ actually depends on the dynamic quantity $m\gamma$, not $m_o$, $\tilde{a}_o \neq a_o$ away from resonance and iteration may have to be used to compute $\xi$ such that a self-consistent solution to (214) is obtained. Secondly, $\nu_e$ can't be zero at $\xi = n$. This is seen (and justified in Appendix B) by adding the non-relativistic kinetic energy $T = \frac{1}{2}h\nu_e$ to the potential energy $V$ evaluated at $\xi = n$ using (210) and (206) together with resonance condition $\nu_e = \nu'$, yielding $-\frac{1}{2}h\nu'$, which is in exact agreement to Bohr's and the Schrödinger equation's quantization of energy.

Applying this as a requirement, $\delta = \frac{3}{2}$.

Thirdly, $c_j = \pm 1$ to ensure the following constraints apply for $\xi = n$, $\tilde{a}_o = a_o$ at resonance, with $1 \le i \le n - (l+1) - 1$:

$$ \sum_{j=1}^{n-(l+1)} \mathbf{Z}_j(n) = \frac{n - (l+1)}{n} \quad (227) $$

$$ \sum_{j=1}^{n-(l+1)} \frac{d |\mathbf{Z}_j(n)|}{d\xi^i} = \frac{d}{d\xi^{i-1}} \left[ \sum_{j=1}^{n-(l+1)} \frac{\operatorname{sgn}(\mathbf{Z}_j(n))}{\xi} + \frac{n - (l+1)}{\xi^2} \right] \Big|_{\xi=n} \quad (228) $$

where (227) is obtained from (223) by enforcing $\nu = \nu'$ at $\xi = n$. The first RHS term of (228) ensures the pole contributions match the same behavior as solutions for $l=n-1, n-2$ that were generated with basic assumptions, whereas $n - (l+1)$ in the second RHS term of (228) corrects for the introduction of higher order derivatives in the reduction of the order of the zero at the origin in (223) with increasing $l$. This needs to happen because the reduction of the zero should truncate terms that appear in higher order derivatives rather than add terms.

The prior analysis is now re-examined, starting with the ground state $n=1, l=0$ and $\nu = 2\nu' (\frac{3}{2} - \frac{1}{\xi})$. As the constraints don't create equations to solve in this case, the pole at the origin is removable, as it originates solely from the solid angle integration that was mapped into $P(\xi)$. This suggests that the correct solution is the one that can remove this second-order pole from the equation and append its associated zero to the solution, just as is done with Schrödinger's solutions. This yields:

$$ P_{1,0}(\xi) = N r^2 e^{-3\xi} = N (\tilde{a}_o \xi)^2 e^{-3\xi} \quad (229) $$

Since the solution to (214) has its integration constant set by $\lambda = 2\pi a_o$ at the Bohr radius, where also by (206), $\xi = 1$:

$$ \lambda_e(\xi) = 2\pi a_o |2 - 3\xi|^{\frac{1}{3}} \quad (230) $$

Applying (6) and (214) with iteration can be used to self-consistently compute $\xi = \frac{r}{a_o}$ for all $r$.

The result is shown in Figure 11, where the 33% shift compared to Schrödinger theory in the probability's
the $cos^2(n\phi)$ probability calculation was observed in [12], it should be noted that observations were the result of a collection of photoelectrons from many atoms, hence the orbits represent a statistical average over variances in phase offsets. Also, virtual forces scramble the $\theta$, $\phi$ dependence of the angular momentum/probability over time, and an applied electric field will have an impact.

To solve for the two variables in (214), an observation from Schrödinger theory is used: The solution for $P(\xi)$ inside the Coulomb potential well for a given energy, with similarity to the square well solution in terms of the generation of nodes, will have unique degenerate solutions with (0,1,...,n-1) zeros at or away from the origin denoted by quantum number $l$, and a 2nd order zero at the origin attributed to $r^2$ from integration over the solid angle of Schrödinger's probability: $P(\xi) = r^2 \int \psi^* \psi d\Omega$. Given the form of (212), this means that the zeros of $P(\xi)$ will appear as poles of $\nu_e(\xi)$.

Thus, considering what is known so far, including also the factor of 2 mentioned previously, $\nu_e$ takes the following general form with constants $\delta$, $c_j$, and original zero $s_j$ of $P(r)$:

$$ \nu_e(\xi) = 2\nu' \left( \delta - \frac{(l+1)}{\xi} - \sum_{j=1}^{n-(l+1)} \mathbf{Z}_j \right) \quad (223) $$

where $2^{l+1}/\xi$ is due to the zero of $P(\xi)$ at the origin with

$$ \mathbf{Z}_j \equiv \frac{1}{c_j \xi - \frac{a_o}{a_o} s_j} \quad (224) $$

representing the poles of $\nu$ away from the origin and $a_o/a_o$ is introduced when replacing $r/a_o$ by $\xi$.

Specific cases are now examined, starting with $n, l = n-1$ where $\nu = 2\nu'(\delta - \frac{n}{\xi})$. Hypothetically, if $\delta = 1$, $\tilde{a}_o = a_o$, and $\xi = \frac{r}{na_o} = n$ at the resonance radii of Bohr's analysis, (212) generates the probability:

$$ P_{n,n-1}(\xi) = N^2 \xi^{2n} e^{-2\xi} \quad (225) $$

which in absorbing a factor of $n$ into the normalization agrees with the square of solutions to Schrödinger's radial equation multiplied by $r^2$. The next case is $n, l = n-2$ in which (223) has form $\nu_e = 2\nu'(1 - \frac{n-1}{\xi} - \frac{1}{x_1\xi - s_1})$ using the same hypothesis as before. Setting $\nu_e = 0$ at $\xi = n$ to be in line with the $n, l = n-1$ case, $c_1 = n$ and $s_1 = n^2 - n$. Solving (212):

$$ P_{n,n-2}(\xi) = N^2 \xi^{2(n-1)} [n\xi - (n^2 - n)]^2 e^{-2\xi} \quad (226) $$

again, matching the solutions to Schrödinger's radial equation multiplied by $r^2$ with a factor of $n$ absorbed by the normalization. This suggests, in principle, boundary conditions can be chosen to enforce agreement with Schrödinger's formulation, however there would be contradictions.

First, as $\xi$ actually depends on the dynamic quantity $m\gamma$, not $m_o$, $\tilde{a}_o \neq a_o$ away from resonance and iteration may have to be used to compute $\xi$ such that a self-consistent solution to (214) is obtained. Secondly, $\nu_e$ can't be zero at $\xi = n$. This is seen (and justified in Appendix B) by adding the non-relativistic kinetic energy $T = \frac{1}{2}h\nu_e$ to the potential energy $V$ evaluated at $\xi = n$ using (210) and (206) together with resonance condition $\nu_e = \nu'$, yielding $-\frac{1}{2}h\nu'$, which is in exact agreement to Bohr's and the Schrödinger equation's quantization of energy.

Applying this as a requirement, $\delta = \frac{3}{2}$.

Thirdly, $c_j = \pm 1$ to ensure the following constraints apply for $\xi = n$, $\tilde{a}_o = a_o$ at resonance, with $1 \le i \le n - (l+1) - 1$:

$$ \sum_{j=1}^{n-(l+1)} \mathbf{Z}_j(n) = \frac{n - (l+1)}{n} \quad (227) $$

$$ \sum_{j=1}^{n-(l+1)} \frac{d|\mathbf{Z}_j(n)|}{d\xi^i} = \frac{d}{d\xi^{i-1}} \left[ \sum_{j=1}^{n-(l+1)} \frac{\operatorname{sgn}(\mathbf{Z}_j(n))}{\xi} + \frac{n-(l+1)}{\xi^2} \right] \Big|_{\xi=n} \quad (228) $$

where (227) is obtained from (223) by enforcing $\nu = \nu'$ at $\xi = n$. The first RHS term of (228) ensures the pole contributions match the same behavior as solutions for $l=n-1, n-2$ that were generated with basic assumptions, whereas $n - (l + 1)$ in the second RHS term of (228) corrects for the introduction of higher order derivatives in the reduction of the order of the zero at the origin in (223) with increasing $l$. This needs to happen because the reduction of the zero should truncate terms that appear in higher order derivatives rather than add terms.

The prior analysis is now re-examined, starting with the ground state $n = 1, l = 0$ and $\nu = 2\nu' (\frac{3}{2} - \frac{1}{\xi})$. As the constraints don't create equations to solve in this case, the pole at the origin is removable, as it originates solely from the solid angle integration that was mapped into $P(\xi)$. This suggests that the correct solution is the one that can remove this second-order pole from the equation and append its associated zero to the solution, just as is done with Schrödinger's solutions. This yields:

$$ P_{1,0}(\xi) = Nr^2e^{-3\xi} = N(\tilde{a}_o\xi)^2e^{-3\xi} \quad (229) $$

Since the solution to (214) has its integration constant set by $\lambda = 2\pi a_o$ at the Bohr radius, where also by (206), $\xi = 1$:

$$ \lambda_e(\xi) = 2\pi a_o |2 - 3\xi|^{\frac{1}{3}} \quad (230) $$

Applying (6) and (214) with iteration can be used to self-consistently compute $\xi = \frac{r}{a_o}$ for all $r$.

The result is shown in Figure 11, where the 33% shift compared to Schrödinger theory in the probability's
<figure><img src="image_13.png" /><figcaption>FIG. 11. Replicating the observed disagreement for the ground state probability distribution between conventional theory and experiment [12] using the variance of quantum mass m and its impact on a self-consistent representation for ξ along with a resonance requirement that specifies a nonzero slope and mγ = m₀ at the Bohr radius (grey dots). Experimental data is not shown here as it was depicted without reference to specific units. The 33% shift of the measured probability's peak compared to conventional theory, the divergence of slopes past the peak, and inflection near the origin, however, are closely matched. In addition, the variation of quantum mass compensates for the probability's shift from Schrödinger's theory and its impact on the expectation value calculation for the ground state energy.</figcaption></figure>

peak, its diverging slope from theory past the peak, and noticeable inflection around the origin, features shown but not discussed in [12], is closely replicated. The experiment measured Stark states, but with a small 808 $\frac{V}{cm}$ field, hence the work here suggests that the mismatch in the original analysis can be traced to the incorrect unperturbed ground state that Schrödinger's theory calculates.

It is also worth noting that the shift in the probability distribution measured by experiment generates a different expectation value for the energy with use of the probability distribution according to Schrödinger's calculation, however using the theoretical approach presented here, the correct energy is maintained. This is because with a lower quantum mass beyond the Bohr radius, the excess difference in probability from Schrödinger's analysis weights a lower kinetic energy, whereas below the Bohr radius, the quantum mass is higher and kinetic energy is higher, hence these two differences cancel out to yield the same energy as represented by Bohr's analysis.

Other nodal measurements matched the authors' analysis, and this is also reflected in this work, as for the case of n = 3, l = 0 with

$$ \nu_e = 2\nu' \left( \frac{3}{2} - \frac{3-2}{\xi} - \frac{1}{\xi - \frac{a_o}{a}} s_1 - \frac{1}{\xi - \frac{a_o}{a}} s_2 \right) \quad (231) $$

where the constraints (227) and (228) take the following

form:

$$ \frac{1}{3-s_1} + \frac{1}{3-s_2} = \frac{2}{3} \\ \frac{1}{(3-s_1)^2} + \frac{1}{(3-s_2)^2} = \frac{2}{3} + \frac{2}{9} \quad (232) $$

The resultant zeros $s_1 = 1.902$ and $s_2 = 7.098$ are in agreement with Schrödinger's solution.

A comparison between theory and measurement in [12] corresponding to the n = 4, l = 0 case is now examined, first by solving the following equations:

$$ \frac{1}{4-s_1} - \frac{1}{4-s_2} + \frac{1}{4-s_3} = \frac{3}{4} \\ \frac{1}{(4-s_1)^2} + \frac{1}{(4-s_2)^2} + \frac{1}{(4-s_3)^2} = \frac{1}{4} + \frac{3}{16} \\ \frac{1}{(4-s_1)^3} + \frac{1}{(4-s_2)^3} + \frac{1}{(4-s_3)^3} = \frac{1}{16} + \frac{3}{64} \quad (233) $$

where one of the poles is of opposite sign, hence the sgn function in (228) cancels two pole contributions to the RHS. The equations were solved self-consistently with an adaptive search technique, yielding values of $s_1 \approx 2.2$, $s_2 \approx 7.1$, $s_3 \approx 12$. These zeros determine the solution to (214) with the use of partial fraction decomposition of the integrand:

$$ \frac{\nu'}{\nu\xi} = \frac{e_1}{\xi - z_1} + \frac{e_2}{\xi - z_2} + \frac{e_3}{\xi - z_3} + \frac{e_4}{\xi - z_4} \quad (234) $$

which generates 4 equations for the coefficients $e_k$ that match up according to the powers of $\xi$. The solution to (214) is then a product, due to integration, with $\lambda$ in the following form:

$$ \lambda = 2\pi n a_o \frac{|\xi - z_1|^{e_1} |\xi - z_2|^{e_2} |\xi - z_3|^{e_3} |\xi - z_4|^{e_4}}{|n - z_1|^{e_1} |n - z_2|^{e_2} |n - z_3|^{e_3} |n - z_4|^{e_4}} \quad (235) $$

where n = 4 and normalization with the $n-z_k$ denominator term originates from the Bohr criterion. The zeros $z_k$ of $\lambda$ are obtained by applying the Newton-Raphson technique to find the roots of the polynomial denominator of the integrand from (214), which is:

$$ 3\xi(\xi - s_1)(\xi - s_2)(\xi - s_3) - 2(n-3)(\xi - s_1)(\xi - s_2)(\xi - s_3) - 2\xi(\xi - s_1)(\xi - s_2) - 2\xi(\xi - s_1)(\xi - s_3) - 2\xi(\xi - s_2)(\xi - s_3) = 0 \quad (236) $$

The values for $e_k:z_k$ were found to be (0.614, 0.214, 0.0871, 0.0849) : (0.436, 2.88, 7.89, 12.9).

Figure 12 reveals different possibilities for (235) to arrive at the same value. As $\frac{m\gamma}{m_o}$ was found from adaptive search to below 0.001% error, the spikes may suggest a chaotic variance in m as the electron orbits the nucleus if the conditions are right. To characterize $\lambda$, $\nu$, and $P(r)$ for comparison with on-resonance measurements in [12], the dominant calculated quantum mass at and beyond $n^2a_o$ is used, which also satisfies the resonance constraint, while its trend below $n^2a_o$ is extended, given that
<figure><img src="image_12.png" /><figcaption>FIG. 11. Replicating the observed disagreement for the ground state probability distribution between conventional theory and experiment [12] using the variance of quantum mass m and its impact on a self-consistent representation for ξ along with a resonance requirement that specifies a nonzero slope and mγ = m₀ at the Bohr radius (grey dots). Experimental data is not shown here as it was depicted without reference to specific units. The 33% shift of the measured probability's peak compared to conventional theory, the divergence of slopes past the peak, and inflection near the origin, however, are closely matched. In addition, the variation of quantum mass compensates for the probability's shift from Schrödinger's theory and its impact on the expectation value calculation for the ground state energy.</figcaption></figure>

peak, its diverging slope from theory past the peak, and noticeable inflection around the origin, features shown but not discussed in [12], is closely replicated. The experiment measured Stark states, but with a small 808 $\frac{V}{cm}$ field, hence the work here suggests that the mismatch in the original analysis can be traced to the incorrect unperturbed ground state that Schrödinger's theory calculates.

It is also worth noting that the shift in the probability distribution measured by experiment generates a different expectation value for the energy with use of the probability distribution according to Schrödinger's calculation, however using the theoretical approach presented here, the correct energy is maintained. This is because with a lower quantum mass beyond the Bohr radius, the excess difference in probability from Schrödinger's analysis weights a lower kinetic energy, whereas below the Bohr radius, the quantum mass is higher and kinetic energy is higher, hence these two differences cancel out to yield the same energy as represented by Bohr's analysis.

Other nodal measurements matched the authors' analysis, and this is also reflected in this work, as for the case of n = 3, l = 0 with

$$ \nu_e = 2\nu' \left( \frac{3}{2} - \frac{3-2}{\xi} - \frac{1}{\xi - \frac{a_o}{a}} s_1 - \frac{1}{\xi - \frac{a_o}{a}} s_2 \right) \quad (231) $$

where the constraints (227) and (228) take the following

form:

$$ \frac{1}{3-s_1} + \frac{1}{3-s_2} = \frac{2}{3} \\ \frac{1}{(3-s_1)^2} + \frac{1}{(3-s_2)^2} = \frac{2}{3} + \frac{2}{9} \quad (232) $$

The resultant zeros $s_1 = 1.902$ and $s_2 = 7.098$ are in agreement with Schrödinger's solution.

A comparison between theory and measurement in [12] corresponding to the n = 4, l = 0 case is now examined, first by solving the following equations:

$$ \frac{1}{4-s_1} - \frac{1}{4-s_2} + \frac{1}{4-s_3} = \frac{3}{4} \\ \frac{1}{(4-s_1)^2} + \frac{1}{(4-s_2)^2} + \frac{1}{(4-s_3)^2} = \frac{1}{4} + \frac{3}{16} \\ \frac{1}{(4-s_1)^3} + \frac{1}{(4-s_2)^3} + \frac{1}{(4-s_3)^3} = \frac{1}{16} + \frac{3}{64} \quad (233) $$

where one of the poles is of opposite sign, hence the sgn function in (228) cancels two pole contributions to the RHS. The equations were solved self-consistently with an adaptive search technique, yielding values of $s_1 \approx 2.2$, $s_2 \approx 7.1$, $s_3 \approx 12$. These zeros determine the solution to (214) with the use of partial fraction decomposition of the integrand:

$$ \frac{\nu'}{\nu\xi} = \frac{e_1}{\xi - z_1} + \frac{e_2}{\xi - z_2} + \frac{e_3}{\xi - z_3} + \frac{e_4}{\xi - z_4} \quad (234) $$

which generates 4 equations for the coefficients $e_k$ that match up according to the powers of $\xi$. The solution to (214) is then a product, due to integration, with $\lambda$ in the following form:

$$ \lambda = 2\pi n a_o \frac{|\xi - z_1|^{e_1} |\xi - z_2|^{e_2} |\xi - z_3|^{e_3} |\xi - z_4|^{e_4}}{|n - z_1|^{e_1} |n - z_2|^{e_2} |n - z_3|^{e_3} |n - z_4|^{e_4}} \quad (235) $$

where n = 4 and normalization with the $n-z_k$ denominator term originates from the Bohr criterion. The zeros $z_k$ of $\lambda$ are obtained by applying the Newton-Raphson technique to find the roots of the polynomial denominator of the integrand from (214), which is:

$$ 3\xi(\xi - s_1)(\xi - s_2)(\xi - s_3) - 2(n-3)(\xi - s_1)(\xi - s_2)(\xi - s_3) - 2\xi(\xi - s_1)(\xi - s_2) - 2\xi(\xi - s_1)(\xi - s_3) - 2\xi(\xi - s_2)(\xi - s_3) = 0 \quad (236) $$

The values for $e_k:z_k$ were found to be (0.614, 0.214, 0.0871, 0.0849) : (0.436, 2.88, 7.89, 12.9).

Figure 12 reveals different possibilities for (235) to arrive at the same value. As $\frac{m\gamma}{m_o}$ was found from adaptive search to below 0.001% error, the spikes may suggest a chaotic variance in m as the electron orbits the nucleus if the conditions are right. To characterize $\lambda$, $\nu$, and $P(r)$ for comparison with on-resonance measurements in [12], the dominant calculated quantum mass at and beyond $n^2a_o$ is used, which also satisfies the resonance constraint, while its trend below $n^2a_o$ is extended, given that
<figure><img src="image_14.png" /><figcaption>FIG. 12. The first graph displays multiple trendlines for the quantum mass of the $n = 4, l = 0$ solution to within 0.001% error with significant instability below the first node (inset), suggesting the electron's chaotic motion. The dominant solution above $n^2 a_o$ (shown in black) satisfies the resonance constraint and is extended below $n^2 a_o$ to calculate a self-consistent solution for the radial probability distribution. The second graph replicates the measured variation of node widths and locations [12] in comparison to Schrödinger's solution, including alignment for the lowest peaks seen in experiment (inset) along with a 20% shift of the highest node. The observed crossover beyond the last peak is also replicated.</figcaption></figure>

small constant perturbations to $\hbar/\lambda^2$, permissible solutions to (213), cause large instability to $m\gamma/m_o$ close to the nucleus.

The plot for $P(r)$ has several features observed experimentally [12], including the substantial narrowing of nodes closest to the nucleus, which together with phase noise explains the bleeding of density measurements there; the alignment to Schrödinger's solution for the two lowest peaks with noticeable shifts in the other two peaks and a 20% shift of the outermost node, plus crossover above Schrödinger's solution past the outermost peak which, unlike Schrödinger's characterization of infinitely bound states, leads to the electron's gain of virtual/background energy and escape at a large distance from the nucleus.

The value of $N$ was chosen to illustrate that the electron spends significantly more time closer to the nucleus compared with Schrödinger's solution. Experimental data displays a high intensity for the lowest peak in contrast to the furthest peak, which is not evident in the radial distribution calculated by the authors [12], suggesting a closer look at the determination of this concentration both on the experimental side and theoretical side is warranted. Given the instability below the first node, it is possible that the computed probability should be lower due to a lower median quantum mass below $m_o$, whereas experimentally there may be limiting factors in determining the sampling of photoelectron energies and concentrations.

## XII. DERIVING DIRAC'S ENERGY LEVELS IN HYDROGEN

As a detailed model for spin is not considered in this work, a different approach utilizing semi-classical reasoning is taken here to derive the energy levels of an electron in hydrogen in parallel with the analysis of Dirac. To begin with, for a relativistic treatment, the principal quantum number $n$ is introduced into the quantization of total energy:

$$E = nh\nu \quad (237)$$

A rotational frame of reference is then chosen for the electron in which the mass-energy precisely cancels the potential energy:

$$m_o c^2 = \frac{Z k_q q^2}{r_b} \quad (238)$$

The radius when this happens is given by:

$$r_b = \frac{Z k_q q^2}{m_o c^2} = \frac{Z \alpha \hbar}{m_o c} \quad (239)$$

where $\alpha$ is the fine structure constant, $\frac{k_q q^2}{\hbar c}$. This criterion takes the place of the Bohr model, i.e. in the relativistic version of that analysis, the motion of the electron at radius $r_b$ is also purely rotational, however one can take advantage of the fact that the centrifugal force acts like a gravitational field. With (238) canceling out the potential energy contribution, the following well-known result for the energy of a rotating particle subjected to the fictitious gravitational field is:

$$E = m_o c^2 \sqrt{1 - \frac{\omega^2 r_b^2}{c^2}} \quad (240)$$

where $\omega = 2\pi\nu$ is the rotational frequency.

As (238) absorbs the electron's rest mass energy, technically it should be subtracted from (240) as a result, which would set $\nu = \nu_e$. However, since $E$ can be defined to within an arbitrary constant when considering absorption or emission phenomenon, this factor is ignored. Substituting (239) into (240) and using $\nu = \frac{E}{nh\nu}$, the following final equation for the energy is obtained:

$$E = m_o c^2 \sqrt{1 - \frac{Z^2 \alpha^2 E^2}{n^2 m_o^2 c^4}} \quad (241)$$

This has the following solution:

$$E = \frac{m_o c^2}{\sqrt{1 + \frac{Z^2 \alpha^2}{n^2}}} \quad (242)$$

Since mass and energy are interchangeable, and the electron's energy at radius $r_b$ is purely rotational with angular momentum $L_\theta = p_\theta r_b$, the following balance equation is obtained with the relativistic formula for energy:

$$ \begin{aligned} E^2 &= p_\theta^2 c^2 + (m_o c^2)^2 \\ &= \frac{(n - n_r)^2 \hbar^2 c^2}{r_b^2} + m_o^2 c^4 = \frac{(j + s)^2 \hbar^2 c^2}{r_b^2} \end{aligned} \quad (243) $$
<figure><img src="image_13.png" /><figcaption>FIG. 12. The first graph displays multiple trendlines for the quantum mass of the $n = 4, l = 0$ solution to within 0.001% error with significant instability below the first node (inset), suggesting the electron's chaotic motion. The dominant solution above $n^2 a_o$ (shown in black) satisfies the resonance constraint and is extended below $n^2 a_o$ to calculate a self-consistent solution for the radial probability distribution. The second graph replicates the measured variation of node widths and locations [12] in comparison to Schrödinger's solution, including alignment for the lowest peaks seen in experiment (inset) along with a 20% shift of the highest node. The observed crossover beyond the last peak is also replicated.</figcaption></figure>

small constant perturbations to $\hbar/\lambda^2$, permissible solutions to (213), cause large instability to $m\gamma/m_o$ close to the nucleus.

The plot for $P(r)$ has several features observed experimentally [12], including the substantial narrowing of nodes closest to the nucleus, which together with phase noise explains the bleeding of density measurements there; the alignment to Schrödinger's solution for the two lowest peaks with noticeable shifts in the other two peaks and a 20% shift of the outermost node, plus crossover above Schrödinger's solution past the outermost peak which, unlike Schrödinger's characterization of infinitely bound states, leads to the electron's gain of virtual/background energy and escape at a large distance from the nucleus.

The value of $N$ was chosen to illustrate that the electron spends significantly more time closer to the nucleus compared with Schrödinger's solution. Experimental data displays a high intensity for the lowest peak in contrast to the furthest peak, which is not evident in the radial distribution calculated by the authors [12], suggesting a closer look at the determination of this concentration both on the experimental side and theoretical side is warranted. Given the instability below the first node, it is possible that the computed probability should be lower due to a lower median quantum mass below $m_o$, whereas experimentally there may be limiting factors in determining the sampling of photoelectron energies and concentrations.

## XII. DERIVING DIRAC'S ENERGY LEVELS IN HYDROGEN

As a detailed model for spin is not considered in this work, a different approach utilizing semi-classical reasoning is taken here to derive the energy levels of an electron in hydrogen in parallel with the analysis of Dirac. To begin with, for a relativistic treatment, the principal quantum number $n$ is introduced into the quantization of total energy:

$$E = nh\nu \quad (237)$$

A rotational frame of reference is then chosen for the electron in which the mass-energy precisely cancels the potential energy:

$$m_o c^2 = \frac{Z k_q q^2}{r_b} \quad (238)$$

The radius when this happens is given by:

$$r_b = \frac{Z k_q q^2}{m_o c^2} = \frac{Z \alpha \hbar}{m_o c} \quad (239)$$

where $\alpha$ is the fine structure constant, $\frac{k_q q^2}{\hbar c}$. This criterion takes the place of the Bohr model, i.e. in the relativistic version of that analysis, the motion of the electron at radius $r_b$ is also purely rotational, however one can take advantage of the fact that the centrifugal force acts like a gravitational field. With (238) canceling out the potential energy contribution, the following well-known result for the energy of a rotating particle subjected to the fictitious gravitational field is:

$$E = m_o c^2 \sqrt{1 - \frac{\omega^2 r_b^2}{c^2}} \quad (240)$$

where $\omega = 2\pi\nu$ is the rotational frequency.

As (238) absorbs the electron's rest mass energy, technically it should be subtracted from (240) as a result, which would set $\nu = \nu_e$. However, since $E$ can be defined to within an arbitrary constant when considering absorption or emission phenomenon, this factor is ignored. Substituting (239) into (240) and using $\nu = \frac{E}{nh\nu}$, the following final equation for the energy is obtained:

$$E = m_o c^2 \sqrt{1 - \frac{Z^2 \alpha^2 E^2}{n^2 m_o^2 c^4}} \quad (241)$$

This has the following solution:

$$E = \frac{m_o c^2}{\sqrt{1 + \frac{Z^2 \alpha^2}{n^2}}} \quad (242)$$

Since mass and energy are interchangeable, and the electron's energy at radius $r_b$ is purely rotational with angular momentum $L_\theta = p_\theta r_b$, the following balance equation is obtained with the relativistic formula for energy:

$$ \begin{aligned} E^2 &= p_\theta^2 c^2 + (m_o c^2)^2 \\ &= \frac{(n - n_r)^2 \hbar^2 c^2}{r_b^2} + m_o^2 c^4 = \frac{(j + s)^2 \hbar^2 c^2}{r_b^2} \end{aligned} \quad (243) $$
where $(j + s)\hbar$ represents the quantization of the LHS with generalized angular momentum quantum number $j = l \pm s$, spin $s = \frac{1}{2}$, and $n_r$ as the integer quantum number assigned to the electron's radial component.

Here, the balance equation recognizes that spin contributes to the expression of mass-energy, $m_o c^2$, as justified further in Appendix A, and overall this must be accounted for in quantization. The subtraction of $n_r$ from $n$ removes the radial and potential energy contribution to the calculation, leaving $L_\theta = (n - n_r)\hbar$. Using (238) to replace the rest mass energy in terms of $r_b$, rearranging terms and taking the square root, this equation yields the well-known relationship from Dirac's derivation:

$$n = n_r + \sqrt{(j + s)^2 - Z^2\alpha^2} \quad (244)$$

In general, it is expected that semi-classical reasoning such as this can be found to introduce quantization into the analysis of other quantum systems. The full justification of this assertion is not presented here, however Appendix B suggests the reason.

### XIII. CONCLUSION

The considerations in this paper introduced classical methods of analysis into the study of quantum systems that offers resolutions to well-known paradoxes associated with standard quantum theory. The agreement with experiments for the problems considered, including results not predicted by standard analysis that led to improvements in accuracy, encourages further expansion of the theory's rigor with 3-D spin modeling and extension to time-dependent analysis.

A number of investigations pertaining to a particle's wave characteristics and its mediation with forces supports the finding that a system's behavior on the de Broglie wavelength scale will deviate from classical behavior in a deterministic fashion when phase coherence is preserved. Wavefunction collapse is replaced by continuous transitions, made possible by interactions with the background, and a modified version of Huygen's principle can be used to calculate interference phenomenon in which wavetrains are replaced by the localization of wave-particle energy referred to as a phase bubble.

### XIV. ACKNOWLEDGMENTS

The author would like to thank Diane Powell for encouraging this research along with her consultation and help in reviewing it with colleagues. Also, his sincere appreciation is extended to Dan Sheehan for his feedback, and Jim Thompson for his illustrations and suggestions.

## Appendix A PROPERTY COUPLING

Unlike Schrödinger's analysis with one solution inside the barrier region of a quantum well, two solutions creates a two-fold degeneracy corresponding to two spin states discussed here. They are found by noting a problem with (43) in that the expression for the velocity does not represent its direction consistently after the particle interacts with either the right or left boundary of the well. To fix this, a 1-D two-state spin model is introduced to replace the action of the $\hat{\chi}_\pm$ operator whereby the velocity's direction is a product of the spin's direction and the expression for $v(x)$. Thus, one has the following spin vector assignments for $v_\pm$ inside the well, using (43) with $v_o \equiv \pm\chi\lambda_a v_a \sin(k_e x) \cos(k_e x)$ representing the envelope's chirally-uncompensated contribution to the velocity:

$$v_+(x < 0) = \begin{bmatrix} 1 \\ -1 \end{bmatrix} v_o \qquad v_+(x > 0) = \begin{bmatrix} -1 \\ 1 \end{bmatrix} v_o \\ v_-(x < 0) = - \begin{bmatrix} -1 \\ 1 \end{bmatrix} v_o \qquad v_-(x > 0) = - \begin{bmatrix} 1 \\ -1 \end{bmatrix} v_o$$

whereby the top spin vector row represents a particle traveling to the left, and the bottom row to the right.

The spin, aligned or anti-aligned with the direction of motion, corrects the probability-based velocity calculation, which reverses sign where the quantum mass either peaks or valleys at $x = 0$. This indicates that not only are there underlying dynamics associated with the transfer of energy to and from the mass component found previously in (64), but also the evolution of quantum mass is coupled to the evolution of spin. Furthermore, the exchange of energy between frequency and mass may impact the localization of charge within the phase bubble, possibly suggesting an acceleration-free, radiation-avoiding charge distribution analogous to what a string theory model would have to address.

## Appendix B THE EIGENVALUE AND MEASUREMENT PROBLEMS

Based on the analysis of Section II and Appendix A, properties evolve along with a change of phase even in the absence of net work. It is then natural to ask if the assumption of the eigenvalue problem still holds that certain observables of an eigenstate are measurably conserved over time.

Before answering this, a different interpretation is suggested concerning the negative and positive energy eigenvalues of Dirac's equation for hydrogen. To explain the appearance of negative energy states, Dirac postulated the negative energy sea model. Though this concept has been replaced by modern quantum field theory, here it is suggested that the states with $\pm$ energy are co-existent in the same manner that the chiral operator introduces positive and negative velocities. Hence, the evolution of
where $(j + s)\hbar$ represents the quantization of the LHS with generalized angular momentum quantum number $j = l \pm s$, spin $s = \frac{1}{2}$, and $n_r$ as the integer quantum number assigned to the electron's radial component.

Here, the balance equation recognizes that spin contributes to the expression of mass-energy, $m_o c^2$, as justified further in Appendix A, and overall this must be accounted for in quantization. The subtraction of $n_r$ from $n$ removes the radial and potential energy contribution to the calculation, leaving $L_\theta = (n - n_r)\hbar$. Using (238) to replace the rest mass energy in terms of $r_b$, rearranging terms and taking the square root, this equation yields the well-known relationship from Dirac's derivation:

$$n = n_r + \sqrt{(j + s)^2 - Z^2 \alpha^2} \quad (244)$$

In general, it is expected that semi-classical reasoning such as this can be found to introduce quantization into the analysis of other quantum systems. The full justification of this assertion is not presented here, however Appendix B suggests the reason.

## XIII. CONCLUSION

The considerations in this paper introduced classical methods of analysis into the study of quantum systems that offers resolutions to well-known paradoxes associated with standard quantum theory. The agreement with experiments for the problems considered, including results not predicted by standard analysis that led to improvements in accuracy, encourages further expansion of the theory's rigor with 3-D spin modeling and extension to time-dependent analysis.

A number of investigations pertaining to a particle's wave characteristics and its mediation with forces supports the finding that a system's behavior on the de Broglie wavelength scale will deviate from classical behavior in a deterministic fashion when phase coherence is preserved. Wavefunction collapse is replaced by continuous transitions, made possible by interactions with the background, and a modified version of Huygen's principle can be used to calculate interference phenomenon in which wavetrains are replaced by the localization of wave-particle energy referred to as a phase bubble.

## XIV. ACKNOWLEDGMENTS

The author would like to thank Diane Powell for encouraging this research along with her consultation and help in reviewing it with colleagues. Also, his sincere appreciation is extended to Dan Sheehan for his feedback, and Jim Thompson for his illustrations and suggestions.

# Appendix A PROPERTY COUPLING

Unlike Schrödinger's analysis with one solution inside the barrier region of a quantum well, two solutions creates a two-fold degeneracy corresponding to two spin states discussed here. They are found by noting a problem with (43) in that the expression for the velocity does not represent its direction consistently after the particle interacts with either the right or left boundary of the well. To fix this, a 1-D two-state spin model is introduced to replace the action of the $\hat{\chi}_\pm$ operator whereby the velocity's direction is a product of the spin's direction and the expression for $v(x)$. Thus, one has the following spin vector assignments for $v_\pm$ inside the well, using (43) with $v_o \equiv \pm\chi\lambda_a\nu_a\sin(k_e x)\cos(k_e x)$ representing the envelope's chirally-uncompensated contribution to the velocity:

$$v_+(x < 0) = \begin{bmatrix} 1 \\ -1 \end{bmatrix} v_o \qquad v_+(x > 0) = \begin{bmatrix} -1 \\ 1 \end{bmatrix} v_o \\ v_-(x < 0) = - \begin{bmatrix} -1 \\ 1 \end{bmatrix} v_o \qquad v_-(x > 0) = - \begin{bmatrix} 1 \\ -1 \end{bmatrix} v_o$$

whereby the top spin vector row represents a particle traveling to the left, and the bottom row to the right.

The spin, aligned or anti-aligned with the direction of motion, corrects the probability-based velocity calculation, which reverses sign where the quantum mass either peaks or valleys at $x = 0$. This indicates that not only are there underlying dynamics associated with the transfer of energy to and from the mass component found previously in (64), but also the evolution of quantum mass is coupled to the evolution of spin. Furthermore, the exchange of energy between frequency and mass may impact the localization of charge within the phase bubble, possibly suggesting an acceleration-free, radiation-avoiding charge distribution analogous to what a string theory model would have to address.

# Appendix B THE EIGENVALUE AND MEASUREMENT PROBLEMS

Based on the analysis of Section II and Appendix A, properties evolve along with a change of phase even in the absence of net work. It is then natural to ask if the assumption of the eigenvalue problem still holds that certain observables of an eigenstate are measurably conserved over time.

Before answering this, a different interpretation is suggested concerning the negative and positive energy eigenvalues of Dirac's equation for hydrogen. To explain the appearance of negative energy states, Dirac postulated the negative energy sea model. Though this concept has been replaced by modern quantum field theory, here it is suggested that the states with $\pm$ energy are co-existent in the same manner that the chiral operator introduces positive and negative velocities. Hence, the evolution of
phase changes the potential and quantum mass-energy relationship such that both ± eigenvalues manifest themselves during a phase cycle for the same state. Expressed differently, due to the association of ± in the equations governing dynamics such as (41), there is a mirror symmetry in the interrelationships between state variables that has no impact on perceived dynamics. For example, in (209), changing the sign of $\nu'$, $\nu$, and applying the $\hat{\chi}_{\pm}$ operator on $\lambda$ has no effect on the solution, though the energy of the system would change sign.

The eigenvalue problem, then, can be recast as an expectation that phase mediation or entanglement between a system and observer is stable over time and classically conservative for canonical properties. The negative energy solutions that Dirac obtained are then unnoticeable given the consistency with how phases align between the observer, the environment, and the observed. More specifically, the positive energy states are observed with a 0° phase for the detector, such as represented by classical correspondence in (64).

The spacetime average of observables and their projection for 0° phase would thus be defined as stationary values pinned by classically defined states $\phi_c$ in phase space: those regions for $\lambda$ and $\phi$ at which the changes in sign of the RHS of (204) is unaffected by a sign flip, i.e. when $\nu = \nu'$. These regions correspond to the semi-classical analysis of the Bohr model or the semi-classical treatment of Dirac's analysis presented here. Stated otherwise, averaging over cycles of the phase:

$$ \langle\psi|H|\psi\rangle = \frac{1}{2} (\langle\psi_+|H|\psi_+\rangle + \langle\psi_-|H|\psi_-\rangle) = \langle\psi_c|H|\psi_c\rangle \quad (245) $$

where $H$ is the Hamiltonian and $|\psi_{\pm}\rangle$ are the phase-swapped, ± representations of $H|\psi\rangle$, which are Schrödinger/Dirac energy eigenstates for hydrogen approximately in agreement with phase-mediated solutions.

## Appendix C BOSE-FERMI STATISTICS AND THE MEDIATION OF PHASE

Pauli's exclusion principle specifies that fermions do not occupy the same eigenstate. Expressed mathematically, a two-particle wavefunction, for example, obeys the following symmetry requirement:

$$ \psi(x_1, x_2) = -\psi(x_2, x_1) \quad (246) $$

where $x_1$ and $x_2$ represent the positions of the two particles. For bosons, such an exchange does not change $\psi$ and overall these considerations are different from classical analysis in which particles are distinguishable.

Both Heisenberg [41] and Dirac [42] are known for their work in characterizing the effects of this exchange property for phenomenon such as ferromagnetism, which Dirac modeled with a term in the Hamiltonian involving the coupling between spins:

$$ H = -J \sum_{<i,j>} \mathbf{S}_i \cdot \mathbf{S}_j \quad (247) $$

Here, electromagnetism determines the coupling constant $J$. While this is best-known for its applications to studies of solids, the derivation of Bose-Fermi statistics does not make reference to an explicit interaction for gases.

Hence, there are those in the scientific community that see the exchange force as fictitious, i.e. the result of considering only the exclusion of wavefunction symmetry in the fermion case, and absence of such exclusion in the boson case [43]. For this theoretical framework, it must then be shown how a weak interaction can have such a profound impact on the statistics of identical particles.

One can look at the experiments conducted with the 1-D Tonks-Girardeau gas [44] whereby a system of bosons can be coerced into fermionic behavior, and vice-versa for other systems. The conclusion here is that given the right type of interactions, it's possible to have Pauli exclusion or bosonic preference for the same state regardless of the particle's spin, because the overall driver involves the reduction of excess energy in the system and stability. This same driver must be present in systems that demonstrate conventional Bose-Fermi statistics. Consider two spin particles that are initially aligned as shown in Figure 13. Which one decides to flip, and which one decides to stay put? For fermions, it is almost like a game of chicken; a contest that requires time to settle, which involves random variables. In the case of bosons, however, especially ones formed from aggregates of fermions, their responses are identical, and that is the problem.

It is postulated that the initial response to repulsion for a spin-1 particle, for example, is the induction of a spin-$\frac{1}{2}$ component that opposes its other spin-$\frac{1}{2}$ component, but this is just a transient state until a complete flip has occurred. Such transients are natural fluctuations in a sea of bosons, and as such they can reduce the magnetic field between two nearby bosons with their spins aligned, in much the same manner that intermediary charges induce attractions between water molecules [45].

As Figure 13 illustrates, bosons with more transient states than aligned states are slightly elevated in energy and unstable, hence they return to a configuration dominated by spin alignment. It is postulated that groups of bosons are mediated by a relatively scant number of transients with the assistance of entanglement as discussed in Section VI. Here, the entanglement ensures that the bosons' phases have a preference to stay aligned with $\langle\phi_F|\mathbf{S}_i \cdot \mathbf{S}_j|\phi\rangle > 0$ remaining relatively stable over time. Fermions, meanwhile, can't split their spin evenly to counter outside forces, so flipping is an all-or-nothing proposition. In this scenario, the dynamics of spin and its impact on the phase favors $\langle\phi_F|\mathbf{S}_i \cdot \mathbf{S}_j|\phi\rangle < 0$, i.e. that the spins are anti-aligned.

It then has to be asked why bosons can't circumvent the impact of transients and anti-align without coercion. As presented in the body of this paper and appendices, a particle is obedient to differential equations that associate a sign, expressed as a 0° or 180° phase, to either the wavelength or the values of $\phi'_F$ for each of its interactions. With fermions, anti-alignment is necessary as otherwise,
phase changes the potential and quantum mass-energy relationship such that both ± eigenvalues manifest themselves during a phase cycle for the same state. Expressed differently, due to the association of ± in the equations governing dynamics such as (41), there is a mirror symmetry in the interrelationships between state variables that has no impact on perceived dynamics. For example, in (209), changing the sign of $v'$, $v$, and applying the $\hat{\chi}_{\pm}$ operator on $\lambda$ has no effect on the solution, though the energy of the system would change sign.

The eigenvalue problem, then, can be recast as an expectation that phase mediation or entanglement between a system and observer is stable over time and classically conservative for canonical properties. The negative energy solutions that Dirac obtained are then unnoticeable given the consistency with how phases align between the observer, the environment, and the observed. More specifically, the positive energy states are observed with a 0° phase for the detector, such as represented by classical correspondence in (64).

The spacetime average of observables and their projection for 0° phase would thus be defined as stationary values pinned by classically defined states $\phi_c$ in phase space: those regions for $\lambda$ and $\phi$ at which the changes in sign of the RHS of (204) is unaffected by a sign flip, i.e. when $v = v'$. These regions correspond to the semi-classical analysis of the Bohr model or the semi-classical treatment of Dirac's analysis presented here. Stated otherwise, averaging over cycles of the phase:

$$ \langle\psi|H|\psi\rangle = \frac{1}{2} (\langle\psi_+|H|\psi_+\rangle + \langle\psi_-|H|\psi_-\rangle) = \langle\psi_c|H|\psi_c\rangle \quad (245) $$

where $H$ is the Hamiltonian and $|\psi_{\pm}\rangle$ are the phase-swapped, ± representations of $H|\psi\rangle$, which are Schrödinger/Dirac energy eigenstates for hydrogen approximately in agreement with phase-mediated solutions.

## Appendix C BOSE-FERMI STATISTICS AND THE MEDIATION OF PHASE

Pauli's exclusion principle specifies that fermions do not occupy the same eigenstate. Expressed mathematically, a two-particle wavefunction, for example, obeys the following symmetry requirement:

$$ \psi(x_1, x_2) = -\psi(x_2, x_1) \quad (246) $$

where $x_1$ and $x_2$ represent the positions of the two particles. For bosons, such an exchange does not change $\psi$ and overall these considerations are different from classical analysis in which particles are distinguishable.

Both Heisenberg [41] and Dirac [42] are known for their work in characterizing the effects of this exchange property for phenomenon such as ferromagnetism, which Dirac modeled with a term in the Hamiltonian involving the coupling between spins:

$$ H = -J \sum_{<i,j>} \mathbf{S}_i \cdot \mathbf{S}_j \quad (247) $$

Here, electromagnetism determines the coupling constant $J$. While this is best-known for its applications to studies of solids, the derivation of Bose-Fermi statistics does not make reference to an explicit interaction for gases.

Hence, there are those in the scientific community that see the exchange force as fictitious, i.e. the result of considering only the exclusion of wavefunction symmetry in the fermion case, and absence of such exclusion in the boson case [43]. For this theoretical framework, it must then be shown how a weak interaction can have such a profound impact on the statistics of identical particles.

One can look at the experiments conducted with the 1-D Tonks-Girardeau gas [44] whereby a system of bosons can be coerced into fermionic behavior, and vice-versa for other systems. The conclusion here is that given the right type of interactions, it's possible to have Pauli exclusion or bosonic preference for the same state regardless of the particle's spin, because the overall driver involves the reduction of excess energy in the system and stability. This same driver must be present in systems that demonstrate conventional Bose-Fermi statistics. Consider two spin particles that are initially aligned as shown in Figure 13. Which one decides to flip, and which one decides to stay put? For fermions, it is almost like a game of chicken; a contest that requires time to settle, which involves random variables. In the case of bosons, however, especially ones formed from aggregates of fermions, their responses are identical, and that is the problem.

It is postulated that the initial response to repulsion for a spin-1 particle, for example, is the induction of a spin-$\frac{1}{2}$ component that opposes its other spin-$\frac{1}{2}$ component, but this is just a transient state until a complete flip has occurred. Such transients are natural fluctuations in a sea of bosons, and as such they can reduce the magnetic field between two nearby bosons with their spins aligned, in much the same manner that intermediary charges induce attractions between water molecules [45].

As Figure 13 illustrates, bosons with more transient states than aligned states are slightly elevated in energy and unstable, hence they return to a configuration dominated by spin alignment. It is postulated that groups of bosons are mediated by a relatively scant number of transients with the assistance of entanglement as discussed in Section VI. Here, the entanglement ensures that the bosons' phases have a preference to stay aligned with $\langle\phi_F|\mathbf{S}_i \cdot \mathbf{S}_j|\phi\rangle > 0$ remaining relatively stable over time. Fermions, meanwhile, can't split their spin evenly to counter outside forces, so flipping is an all-or-nothing proposition. In this scenario, the dynamics of spin and its impact on the phase favors $\langle\phi_F|\mathbf{S}_i \cdot \mathbf{S}_j|\phi\rangle < 0$, i.e. that the spins are anti-aligned.

It then has to be asked why bosons can't circumvent the impact of transients and anti-align without coercion. As presented in the body of this paper and appendices, a particle is obedient to differential equations that associate a sign, expressed as a 0° or 180° phase, to either the wavelength or the values of $\phi'_F$ for each of its interactions. With fermions, anti-alignment is necessary as otherwise,
<figure><img src="image_15.png" /><figcaption>FIG. 13. Intermediate states associated with the interaction of bosons and fermions. Fermions are able to seek the lowest energy state and remain stable with exclusion while bosons require transient states and entanglement to reduce the interaction energy.</figcaption></figure>

their random interactions would create n-body chaos in place of phase entanglement, leading to the destruction of the stability of quantum states. For bosons, this option isn't available due to an intermediate step during engagement based on a symmetry argument. Once transients form, they bias the phase mediation process of nearby bosons during entanglement, which ensures that the system seeks the lowest possible energy state by reducing some but not all of the number of transients.

In conclusion, Bose-Fermi statistics is the result of energetically favored interactions enhanced by the entanglement that occurs with phase mediation, as opposed to attributing the phenomenon to a phantom, interaction-free allowance or forbiddance of wavefunction overlap.

[1] E. Schrödinger, Quantisation as a problem of proper values (part i), Annalen der Physik **79**, 1 (1926).

[2] T. E. Hartman, Tunneling of a wave packet, J. Appl. Phys. **33**, 3427 (1962).

[3] R. Ramós, D. Spierings, I. Racicot, and A. M. Steinberg, Measurement of the time spent by a tunnelling atom within the barrier region, Nature **583**, 529 (2020).

[4] G. C. Ghirardi, A. Rimini, and T. Weber, Unified dynamics for microscopic and macroscopic systems, Phys. Rev. D **34**, 470 (1986).

[5] P. Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A **39**, 2277 (1989).

[6] I. J. A. et al., Search for spontaneous radiation from wave function collapse in the majorana demonstrator, Phys. Rev. Lett. **129**, 080401 (2022).

[7] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, Wave-particle duality of $c_{60}$ molecules, Nature **401**, 680 (1999).

[8] M. F. Pusey, J. Barrett, and T. Rudolph, On the reality of the quantum state, Nature Physics **8**, 475 (2012).

[9] *Journal of Physics: Conf. Ser.*, Vol. 442 (2013).

[10] J. S. Bell, *Speakable and Unspeakable in Quantum Mechanics* (Cambridge University Press, Cambridge, England, 2004).

[11] T. Maudlin, *Philosophy of Physics: Quantum Theory*, Vol. 33 (Princeton University Press, Princeton, New Jersey, 2019).

[12] A. S. Stodolna, A. Rouzée, F. Lèpine, S. Cohen, F. Robicheaux, and et al., Hydrogen atoms under magnification: Direct observation of the nodal structure of stark states, Phys. Rev. Lett. **110**, 213001 (2013).

[13] T. Prosen, Chaos and complexity of quantum motion, J. Phys. A **40**, 7881 (2007).

[14] T. Ali, A. Bhattacharyya, and S. S. H. et al., Chaos and complexity in quantum mechanics, Phys. Rev. D **101**, 026021 (2020).

[15] A. Einstein, The photoelectric effect, Annalen der Physik **17–18** (1905).

[16] L. de Broglie, *Recherches sur la Théorie des Quanta*, Ph.D. thesis, Masson, Paris (1924).

[17] P. Dirac, *Albert Einstein, Historical and Cultural Perspectives: The Centennial Symposium in Jerusalem*, edited by G. J. Holton and Y. Elkana (Princeton University Press, Princeton, New Jersey, 1982) pp. 79–90.

[18] I. Newton, *Newton's Principia. The mathematical principles of natural philosophy* (D. Adee, New York, 1846).

[19] M. Renou, D. Trillo, and M. W. et al., Quantum theory based on real numbers can be experimentally falsified, Nature **600**, 625 (2021).

[20] A. Montina, Exponential complexity and ontological theories of quantum mechanics, Phys. Rev. A **77**, 022104 (2008).

[21] J. R. Hance and S. Hossenfelder, The wave function as a true ensemble, Proc. R. Soc. A **478**, 20210705 (2022).

[22] K. Wang and Y. Zhong, Frequency modulation of gravitational waves by ultralight scalar dark matter (2023), arXiv:2306.10732 [astro-ph.CO].

[23] J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika **1**, 195 (1964).

[24] P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett. **13**, 508 (1964).

[25] E. Stueckelberg, La signification du temps propre en mécanique ondulatoire, Helv. Phys. Acta **14**, 322 (1941).

[26] R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. **20**, 367 (1948).

[27] H. Collaboration, C. Adloff, V. Andreev, and B. A. et. al., Search for compositeness, leptoquarks and large extra dimensions in eq contact interactions at hera, Physics Letters B **479**, 358 (2000).

[28] M. Koberidze, A. V. Feshchenko, M. J. Puska, R. M. Nieminen, and J. P. Pekola, Effect of interface geometry on electron tunnelling in al/al2o3/al junctions, J. Phys. D **49**, 165303 (2016).

[29] J. Liu, X. Xu, and M. P. Anantram, Role of inelastic electron-phonon scattering in electron transport through ultra-scaled amorphous phase change material nanos-structures, J. Comput Electron **10.1007/s10825-014-0579-7** (2014).

[30] H.-J. Fitting, E. Schreiber, and I. A. Glavatskikh, Monte carlo modeling of electron scattering in nonconductive specimens, Microscopy and Microanalysis **10**, 764 (2004).
<figure><img src="image_14.png" /><figcaption>FIG. 13. Intermediate states associated with the interaction of bosons and fermions. Fermions are able to seek the lowest energy state and remain stable with exclusion while bosons require transient states and entanglement to reduce the interaction energy.</figcaption></figure>

their random interactions would create n-body chaos in place of phase entanglement, leading to the destruction of the stability of quantum states. For bosons, this option isn't available due to an intermediate step during engagement based on a symmetry argument. Once transients form, they bias the phase mediation process of nearby bosons during entanglement, which ensures that the system seeks the lowest possible energy state by reducing some but not all of the number of transients.

In conclusion, Bose-Fermi statistics is the result of energetically favored interactions enhanced by the entanglement that occurs with phase mediation, as opposed to attributing the phenomenon to a phantom, interaction-free allowance or forbiddance of wavefunction overlap.

[1] E. Schrödinger, Quantisation as a problem of proper values (part i), Annalen der Physik **79**, 1 (1926).

[2] T. E. Hartman, Tunneling of a wave packet, J. Appl. Phys. **33**, 3427 (1962).

[3] R. Ramós, D. Spierings, I. Racicot, and A. M. Steinberg, Measurement of the time spent by a tunnelling atom within the barrier region, Nature **583**, 529 (2020).

[4] G. C. Ghirardi, A. Rimini, and T. Weber, Unified dynamics for microscopic and macroscopic systems, Phys. Rev. D **34**, 470 (1986).

[5] P. Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A **39**, 2277 (1989).

[6] I. J. A. et al., Search for spontaneous radiation from wave function collapse in the majorana demonstrator, Phys. Rev. Lett. **129**, 080401 (2022).

[7] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, Wave-particle duality of $c_{60}$ molecules, Nature **401**, 680 (1999).

[8] M. F. Pusey, J. Barrett, and T. Rudolph, On the reality of the quantum state, Nature Physics **8**, 475 (2012).

[9] *Journal of Physics: Conf. Ser.*, Vol. 442 (2013).

[10] J. S. Bell, *Speakable and Unspeakable in Quantum Mechanics* (Cambridge University Press, Cambridge, England, 2004).

[11] T. Maudlin, *Philosophy of Physics: Quantum Theory*, Vol. 33 (Princeton University Press, Princeton, New Jersey, 2019).

[12] A. S. Stodolna, A. Rouzée, F. Lèpine, S. Cohen, F. Robicheaux, and et al., Hydrogen atoms under magnification: Direct observation of the nodal structure of stark states, Phys. Rev. Lett. **110**, 213001 (2013).

[13] T. Prosen, Chaos and complexity of quantum motion, J. Phys. A **40**, 7881 (2007).

[14] T. Ali, A. Bhattacharyya, and S. S. H. et al., Chaos and complexity in quantum mechanics, Phys. Rev. D **101**, 026021 (2020).

[15] A. Einstein, The photoelectric effect, Annalen der Physik **17–18** (1905).

[16] L. de Broglie, *Recherches sur la Théorie des Quanta*, Ph.D. thesis, Masson, Paris (1924).

[17] P. Dirac, *Albert Einstein, Historical and Cultural Perspectives: The Centennial Symposium in Jerusalem*, edited by G. J. Holton and Y. Elkana (Princeton University Press, Princeton, New Jersey, 1982) pp. 79–90.

[18] I. Newton, *Newton's Principia. The mathematical principles of natural philosophy* (D. Adee, New York, 1846).

[19] M. Renou, D. Trillo, and M. W. et al., Quantum theory based on real numbers can be experimentally falsified, Nature **600**, 625 (2021).

[20] A. Montina, Exponential complexity and ontological theories of quantum mechanics, Phys. Rev. A **77**, 022104 (2008).

[21] J. R. Hance and S. Hossenfelder, The wave function as a true ensemble, Proc. R. Soc. A **478**, 20210705 (2022).

[22] K. Wang and Y. Zhong, Frequency modulation of gravitational waves by ultralight scalar dark matter (2023), arXiv:2306.10732 [astro-ph.CO].

[23] J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika **1**, 195 (1964).

[24] P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett. **13**, 508 (1964).

[25] E. Stueckelberg, La signification du temps propre en mécanique ondulatoire, Helv. Phys. Acta **14**, 322 (1941).

[26] R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. **20**, 367 (1948).

[27] H. Collaboration, C. Adloff, V. Andreev, and B. A. et. al., Search for compositeness, leptoquarks and large extra dimensions in eq contact interactions at hera, Physics Letters B **479**, 358 (2000).

[28] M. Koberidze, A. V. Feshchenko, M. J. Puska, R. M. Nieminen, and J. P. Pekola, Effect of interface geometry on electron tunnelling in al/al2o3/al junctions, J. Phys. D **49**, 165303 (2016).

[29] J. Liu, X. Xu, and M. P. Anantram, Role of inelastic electron-phonon scattering in electron transport through ultra-scaled amorphous phase change material nanos-structures, J. Comput Electron **10.1007/s10825-014-0579-7** (2014).

[30] H.-J. Fitting, E. Schreiber, and I. A. Glavatskikh, Monte carlo modeling of electron scattering in nonconductive specimens, Microscopy and Microanalysis **10**, 764 (2004).
[31] K. Flensberg, S. M. Girvin, M. Jonson, D. R. Penn, and M. Stiles, Coulomb blockade in single tunnel-junctions: quantum mechanical effects of the electromagnetic environment, Z. Phys. B - Condensed Matter **85**, 395 (1991).

[32] J. G. Simmons, Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film, J. Appl. Phys. **34**, 1793 (1963).

[33] R. H. Fowler and L. Nordheim, Electron emission in intense electric fields, Proc. R. Soc. A **119**, 173 (1928).

[34] J. D. R. Buchanan, T. P. A. Hase, and B. K. Tanner, Determination of the thickness of al2o3 barriers in magnetic tunnel junctions, Appl. Phys. Lett. **81**, 751 (2002).

[35] B. J. Jönsson-Akerman, R. Escudero, C. Leighton, S. Kim, and et. al., Reliability of normal-state current-voltage characteristics as an indicator of tunnel-junction barrier quality, Appl. Phys. Lett. **77**, 1870 (2000).

[36] H. Goldstein, *Classical Mechanics 2nd Edition* (Addison-Wesley Publishing Company, Boston, Massachusetts, 1981).

[37] E. Nelson, Derivation of the schrödinger equation from newtonian mechanics, Phys. Rev. **150**, 1079 (1966).

[38] G. Grössing, Derivation of the schrödinger equation and the klein-gordon equation from first principles, Preprint

(2002), arXiv:quant-ph/0205047.

[39] E. Noether, Invariante variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Göttingen **1918**, 235 (1918).

[40] N. Bohr, On the constitution of atoms and molecules, Phil. Mag. S. 6. **26**, 24 (1913).

[41] W. Heisenberg, Mehrkörperproblem und resonanz in der quantenmechanik, Zeitschrift für Physik **38**, 411 (1926).

[42] P. A. M. Dirac, On the theory of quantum mechanics, Proc. of Royal Society A **112**, 661 (1926).

[43] W. J. Mullin and G. Blaylock, Quantum statistics: Is there an effective fermion repulsion or boson attraction?, American Journal of Physics **71**, 1223 (2003).

[44] T. Kinoshita, T. Wenger, and D. S. Weiss, Observation of a one-dimensional tonsks-girardeau gas, Science **305**, 1125 (2004).

[45] E. Nagornyak and H. Y. and Gerald H. Pollack, Mechanism of attraction between like-charged particles in aqueous solution, Royal Society of Chemistry: Soft Matter **5**, 10.1039/b905080a (2009).
[31] K. Flensberg, S. M. Girvin, M. Jonson, D. R. Penn, and M. Stiles, Coulomb blockade in single tunnel-junctions: quantum mechanical effects of the electromagnetic environment, Z. Phys. B - Condensed Matter **85**, 395 (1991).

[32] J. G. Simmons, Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film, J. Appl. Phys. **34**, 1793 (1963).

[33] R. H. Fowler and L. Nordheim, Electron emission in intense electric fields, Proc. R. Soc. A **119**, 173 (1928).

[34] J. D. R. Buchanan, T. P. A. Hase, and B. K. Tanner, Determination of the thickness of al2o3 barriers in magnetic tunnel junctions, Appl. Phys. Lett. **81**, 751 (2002).

[35] B. J. Jönsson-Akerman, R. Escudero, C. Leighton, S. Kim, and et. al., Reliability of normal-state current-voltage characteristics as an indicator of tunnel-junction barrier quality, Appl. Phys. Lett. **77**, 1870 (2000).

[36] H. Goldstein, *Classical Mechanics 2nd Edition* (Addison-Wesley Publishing Company, Boston, Massachusetts, 1981).

[37] E. Nelson, Derivation of the schrödinger equation from newtonian mechanics, Phys. Rev. **150**, 1079 (1966).

[38] G. Grössing, Derivation of the schrödinger equation and the klein-gordon equation from first principles, Preprint

(2002), arXiv:quant-ph/0205047.

[39] E. Noether, Invariante variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Göttingen **1918**, 235 (1918).

[40] N. Bohr, On the constitution of atoms and molecules, Phil. Mag. S. 6. **26**, 24 (1913).

[41] W. Heisenberg, Mehrkörperproblem und resonanz in der quantenmechanik, Zeitschrift für Physik **38**, 411 (1926).

[42] P. A. M. Dirac, On the theory of quantum mechanics, Proc. of Royal Society A **112**, 661 (1926).

[43] W. J. Mullin and G. Blaylock, Quantum statistics: Is there an effective fermion repulsion or boson attraction?, American Journal of Physics **71**, 1223 (2003).

[44] T. Kinoshita, T. Wenger, and D. S. Weiss, Observation of a one-dimensional tonsks-girardeau gas, Science **305**, 1125 (2004).

[45] E. Nagornyak and H. Y. and Gerald H. Pollack, Mechanism of attraction between like-charged particles in aqueous solution, Royal Society of Chemistry: Soft Matter **5**, 10.1039/b905080a (2009).