Theoretical and experimental justification for the Schrödinger equation
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles. Classical electromagnetic waves. Nature of light. The quantum particle of light is called a photon. Light has both a wave-like and a particle-like nature. In other words, light can appear to be made of photons (particles) in some experiments and light can  act like waves in other experiments. The dynamics of classical electromagnetic waves are completely described by Maxwell's equations, the classical description of electrodynamics. In the absence of sources, Maxwell's equations can be written as wave equations in the electric and magnetic field vectors. Maxwell's equations thus describe, among other things, the wave-like properties of light. When "classical" (coherent or thermal) light is incident on a photographic plate or CCD, the average number of "hits", "dots", or "clicks" per unit time that result is approximately proportional to the square of the electromagnetic fields of the light. By  formal analogy, the wavefunction of a material particle can be used to find the probability density by taking its absolute-value squared. Unlike electromagnetic fields, quantum-mechanical wavefunctions are complex. (Often in the case of EM fields complex notation is used for convenience, but it is understood that in fact the fields are real. However, wavefunctions are genuinely complex.) Maxwell's equations were completely known by the latter part of the nineteenth century. The dynamical equations for light were, therefore, well-known long before the discovery of the photon. This is not true for other particles such as the electron . It was surmised from the interaction of light with atoms that electrons also had both a particle-like and a wave-like nature. Newtonian mechanics, a description of the particle-like behavior of macroscopic objects, failed to describe very small objects such as electrons. Abductive reasoning was performed to obtain the dynamics of massive objects (particles with mass) such as electrons. The electromagnetic wave equation, the equation that described the dynamics of light, was used as a prototype for discovering the Schrödinger equation, the equation that describes the wave-like and particle-like dynamics of nonrelativistic massive particles. Plane sinusoidal  waves. Electromagnetic wave equation. The electromagnetic wave equation describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form: where "c" is the speed of light in the medium. In a vacuum, c = 2.998 × 10 meters per second, which is the speed of light in free space. The magnetic field is related to the electric field through Faraday's law (cgs units) Plane wave solution of the electromagnetic wave equation. The plane sinusoidal  solution for an electromagnetic wave traveling in the z direction is (cgs units and SI units) formula_4 for the electric field and formula_5 for the magnetic field, where "k" is the wavenumber, formula_6 is the angular frequency of the wave, and formula_7 is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions. In complex notation, the quantity formula_8 is the amplitude of the wave. Here formula_9 is the Jones vector in the x-y plane. The notation for this vector is the bra–ket notation of Dirac, which is normally  used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector. The angles formula_10 are the angle the electric field makes with the x axis and the two initial phases of the wave, respectively. The quantity formula_11 is the state vector of the wave. It describes the polarization of the wave and the spatial and temporal functionality of the wave. For a coherent state light beam so dim that its average photon number is much less than 1, this is approximately equivalent to the quantum state  of a single photon. Energy density of classical electromagnetic waves. Energy in a plane wave. The energy per unit volume in classical electromagnetic fields is (cgs units) For a plane wave, converting to complex notation (and hence dividing by a factor of 2), this becomes where the energy has been averaged over a wavelength of the wave. Fraction of energy in each component. The fraction of energy in the x component of the plane wave (assuming linear polarization) is with a similar expression for the y component. The fraction in both components is Momentum density of classical electromagnetic waves. The  momentum density is given by the Poynting vector For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density: The momentum density has been averaged over a wavelength. Angular momentum density of classical electromagnetic waves. The angular momentum density is For a sinusoidal plane wave the angular momentum is in the z direction and is given by (going over to complex notation) where again the density is averaged over a wavelength. Here right and left circularly polarized unit vectors are defined as and Unitary operators and energy  conservation. A wave can be transformed by, for example, passing through a birefringent crystal or through slits in a diffraction grating. We can define the transformation of the state from the state at time t to the state at time formula_22 as To conserve energy in the wave we require where formula_25 is the adjoint of U, the complex conjugate transpose of the matrix. This implies that a transformation that conserves energy must obey where I is the identity operator and U is called a unitary operator. The unitary property is necessary to ensure energy conservation in state transformations. Hermitian  operators and energy conservation. If formula_27 is an infinitesimal real quantity formula_28, then the unitary transformation is very close to the identity matrix (the final state is very close to the initial state) and can be written and the adjoint by The factor of "i" is introduced for convenience. With this convention, it will be shown that energy conservation requires H to be a Hermitian operator and that H is related to the energy of a particle. Energy conservation requires Since formula_32 is infinitesimal, which means that formula_33 may be neglected with respect to formula_32, the last term can be  omitted. Further, if "H" is equal to its adjoint: it follows that (for infinitesimal translations in time formula_36) so that, indeed, energy is conserved. Operators that are equal to their adjoints are called Hermitian or self-adjoint. The infinitesimal translation of the polarization state is Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator. While this derivation is classical, the concept of a Hermitian operator generating energy-conserving infinitesimal transformations forms an important basis for quantum mechanics. The derivation of the Schrödinger equation follows directly from this concept. Quantum analogy  of classical electrodynamics. The treatment to this point has been classical. However, the quantum mechanical treatment of particles follows along lines "formally analogous" however, to Maxwell's equations for electrodynamics. The analog of the classical "state vectors" formula_39 in the classical description is quantum state vectors in the description of photons. Energy, momentum, and angular momentum of photons. Energy. The early interpretation is based on the experiments of Max Planck and the interpretation of those experiments by Albert Einstein, which was that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is  related to the angular frequency of the wave by the relation formula_40 where formula_41 is an experimentally determined quantity known as the reduced Planck's constant. If there are formula_42 photons in a box of volume formula_43, the energy (neglecting zero point energy) in the electromagnetic field is formula_44 and the energy density is formula_45 The energy of a photon can be related to classical fields through the correspondence principle which states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large formula_46, the quantum energy density must be the same as  the classical energy density formula_47 The average number of photons in the box in a coherent state is then formula_48 Momentum. The correspondence principle also determines the momentum and angular momentum of the photon. For momentum formula_49 which implies that the momentum of a photon is formula_50 (or equivalently formula_51). Angular momentum and spin. Similarly for the angular momentum formula_52 which implies that the angular momentum of the photon is formula_53 the quantum interpretation of this expression is that the photon has a probability of formula_54 of having an angular momentum of formula_41 and a probability of formula_56 of having  an angular momentum of formula_57. We can therefore think of the angular momentum of the photon being quantized as well as the energy. This has indeed been experimentally verified. Photons have only been observed to have angular momenta of formula_58. Spin operator. The spin of the photon is defined as the coefficient of formula_41 in the angular momentum calculation. A photon has spin 1 if it is in the formula_60 state and -1 if it is in the formula_61 state. The spin operator is defined as the outer product formula_62 The eigenvectors of the spin operator are formula_63 and formula_64  with eigenvalues 1 and -1, respectively. The expected value of a spin measurement on a photon is then formula_65 An operator S has been associated with an observable quantity, the angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for angular momentum, but it is in general true for any observable quantity. Probability for a single photon. There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to  calculate the likelihood of a single photon to be in a particular state. The former interpretation is applicable to thermal or to coherent light (see Quantum optics). The latter interpretation is the option for a single-photon Fock state. Dirac explains this in the context of the double-slit experiment: Probability amplitudes. The probability for a photon to be in a particular polarization state depends on the probability distribution over the fields as calculated by the classical Maxwell's equations (in the Glauber-Sudarshan P-representation of a one-photon Fock state.) The expectation value of the photon number in  a coherent state in a limited region of space is quadratic in the fields. In quantum mechanics, by analogy, the state or probability amplitude of a single particle contains the basic probability information. In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities: (The following quote is from Baym, Chapter 1) de Broglie waves. In 1923 Louis de Broglie addressed the question of whether all particles can have both a wave and a particle nature similar to the photon. Photons differ from many other particles in that they are massless and  travel at the speed of light. Specifically de Broglie asked the question of whether a particle that has both a wave and a particle associated with it is consistent with Einstein's two great 1905 contributions, the special theory of relativity and the quantization of energy and momentum. The answer turned out to be positive. The wave and particle nature of electrons was experimentally observed in 1927, two years after the discovery of the Schrödinger equation. de Broglie hypothesis. De Broglie supposed that every particle was associated with both a particle and a wave. The angular frequency formula_66 and wavenumber  formula_67 of the wave was related to the energy E and momentum p of the particle by formula_68 and formula_69 The question reduces to whether every observer in every inertial reference frame can agree on the phase of the wave. If so, then a wave-like description of particles may be consistent with special relativity. Rest frame. First consider the rest frame of the particle. In that case the frequency and wavenumber of the wave are related to the energy and momentum of the particles properties by formula_70 and formula_71 where "m" is the rest mass of the particle. This  describes a wave of infinite wavelength and infinite phase velocity formula_72 The wave may be written as proportional to formula_73 This, however, is also the solution for a simple harmonic oscillator, which can be thought of as a clock in the rest frame of the particle. We can imagine a clock ticking at the same frequency as the wave is oscillating. The phases of the wave and the clock can be synchronized. Frame of the observer. It is shown that the phase of the wave in an observer frame is the same as the phase of the wave in a  particle frame, and also the same as clocks in the two frames. There is, therefore, consistency of both a wave-like and a particle-like picture in special relativity. Phase of the observer clock. In the frame of an observer moving at relative speed v with respect to the particle, the particle clock is observed to tick at a frequency formula_74 where formula_75 is a Lorentz factor that describes time dilation of the particle clock as observed by the observer. The phase of the observer clock is formula_76 where formula_77 is time measured in the particle frame. Both the observer  clock and the particle clock agree on the phase. Phase of the observer wave. The frequency and wavenumber of the wave in the observer frame is given by formula_78 and formula_79 with a phase velocity formula_80 The phase of the wave in the observer frame is formula_81 The phase of the wave in the observer frame is the same as the phase in the particle frame, as the clock in the particle frame, and the clock in the observer frame. A wave-like picture of particles is thus consistent with special relativity. In fact, we now know that these relations  can be succinctly written using special relativistic 4-vector notation: The relevant four-vectors are: The relations between the four-vectors are as follows: The phase of the wave is the relativistic invariant: formula_89 Bohr atom. Inconsistency of observation with classical physics. The de Broglie hypothesis helped resolve outstanding issues in atomic physics. Classical physics was unable to explain the observed behaviour of electrons in atoms. Specifically, accelerating electrons emit electromagnetic radiation according to the Larmor formula. Electrons orbiting a nucleus should lose energy to radiation and eventually spiral into the nucleus. This is not observed. Atoms are stable on  timescales much longer than predicted by the classical Larmor formula. Also, it was noted that excited atoms emit radiation with discrete frequencies. Einstein used this fact to interpret discrete energy packets of light as, in fact, real particles. If these real particles are emitted from atoms in discrete energy packets, however, must the emitters, the electrons, also change energy in discrete energy packets? There is nothing in Newtonian mechanics that explains this. The de Broglie hypothesis helped explain these phenomena by noting that the only allowed states for an electron orbiting an atom are those that allow for standing waves  associated with each electron. Balmer series. The Balmer series identifies those frequencies of light that can be emitted from an excited hydrogen atom: where R is known as the Rydberg constant and is equal to 13.6 electron volts. Assumptions of the Bohr model. The Bohr model, introduced in 1913, was an attempt to provide a theoretical basis for the Balmer series. The assumptions of the model are: Implications of the Bohr model. In a circular orbit the centrifugal force balances the attractive force of the electron where e is the charge on the electron or proton. The energy of the  orbiting electron is