"Electron Orbits and Space Quantization: A Deep Dive into Modern Physics"
Published: 2024-01-16 14:33:28

Here's a mind bending conversation (to my mind at least!) I just had with Claude.ai, and then I put the conversation itself into ChatGPT to get a summary. 

# Background
Ever since I was in school at Michigan earning my Mechanical Engineering degree but also a complete computer nerd at heart I've wondered why quantum theory applies to particles, positions and velocities but stops abruptly at quantization of space itself.  I just had a chat with Claude.ai and it was interesting.  Warning, if you read it you might want to smoke some weed!

Here's the summary from ChatGPT and following that the Claude.AI Chat. Here' the public link to the ChatGPT part.

[ChatGPT public link](https://chat.openai.com/share/262ca27f-b799-461e-9b2f-25fde8f7aefd)

# ChatGPT Summary

This conversation with Claude.ai delves into complex topics regarding the nature of electron motion, the continuous nature of space, and the theoretical considerations of space quantization. It reflects a deep engagement with quantum mechanics and theoretical physics, highlighting key concepts and ongoing debates in these fields.

- Electron Motion and Quantum Mechanics: The conversation begins with a discussion about the nature of electron motion around the nucleus. It explains that while electrons do not have fixed orbits like planets, they exhibit continuous motion within quantized energy levels and orbitals. This is a fundamental aspect of quantum mechanics, where the probabilistic nature of electron positions is described by wavefunctions.
- Quantum Continuity and Discreteness: The discussion progresses to reconcile the seeming contradiction between quantization (discrete energy levels and states) and the continuous motion of electrons. It clarifies that quantum mechanics involves continuous functions for the probability wave evolution, with discontinuities only occurring during measurements or state transitions.
- Nature of Space: The concept of space as a continuous mathematical continuum, even at the quantum level, is affirmed. Despite the quantized behavior of subatomic particles, the mathematical models in physics still treat space as a continuous entity, without minimum length scales like the Planck length playing a discretizing role.
- Quantization of Space: The conversation shifts to theoretical explorations about the possibility of space itself being quantized. It mentions various theoretical frameworks like loop quantum gravity and lattice quantum field theories, which speculate on the discretization of space at the smallest scales. However, it also emphasizes that these are speculative and lack experimental verification.
- Theoretical Implications and Speculations: Towards the end, the conversation considers the broader implications of space quantization, especially in the context of unifying quantum mechanics with general relativity. It acknowledges the theoretical appeal of a quantized spacetime but also highlights the lack of definitive experimental evidence and the continued success of continuous models.
- Philosophical and Mathematical Perspectives: Finally, the conversation touches upon a philosophical perspective, pondering whether the Pythagorean rejection of irrationals and a focus on discretization might align with modern physics' understanding of the universe, especially in light of quantum theories and the limits of the Planck scale.

Overall, this conversation showcases a thoughtful and informed engagement with complex topics at the intersection of quantum mechanics, theoretical physics, and the philosophy of science.

# Claude.AI Conversation

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I understand that we can never pin point the location of an electron that moves about a proton, but do we believe that it has a continuous motion?
JC
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Yes, electrons are generally considered to have continuous motion around the nucleus of an atom. Some key points about the motion of electrons:

- Electrons do not orbit the nucleus in fixed paths like planets. Rather they exist in orbitals or electron clouds that indicate probability regions where the electrons might be found.
- Electron positions and velocities are quantized, meaning they can only have certain discrete values. But between changes in those states, electrons still move in a continuous fashion rather than teleporting between locations.
- Electron motion is very rapid on tiny atomic scales. Even in the ground state, electrons move at speeds of around 106 m/s as they rapidly orbit the nucleus.
- When electrons transition between energy levels/orbitals, they emit or absorb photons in a continuous manner. If their motion was not continuous, quantum mechanics would need to be revised.

So while we cannot know both the exact position and momentum/velocity at the same time due to quantum uncertainty, electrons appear to describe continuous trajectories between interactions and measurements on the atomic scale based on our understanding from the Schrödinger equation solutions. Their probabilistic clouds dynamically evolve in continuous space and time.

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Since electron positions and velocities are quantized how are they able to move in a continuous fashion?
JC
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Excellent question! The quantization of electrons does not mean their motion is discontinuous, even though it seems counterintuitive. Here's a more detailed explanation:

The quantized energy levels and orbitals do define specific states that electrons are allowed to occupy and transitions between them must be discrete jumps. However, the equations of quantum mechanics still involve continuous functions to describe how the probabilities evolve in time and space between measurement events that would collapse the wavefunction.

Specifically, the time-dependent Schrödinger equation predicts smooth continuous evolution of the wavefunction between measurements. And electrons within stationary states still have a wavelength-defined velocity as probability waves even in their ground state.

Additionally, while orbitals define discontinuous shells of fixed energy, their cloud shape morphs smoothly in time and space and the electrons can occupy any point within it with a defined probability density.

So in summary - the discontinuities come from measurement intersections with allowed states. But probability waves in between evolve smoothly across both position and momentum variables. The underlying wave mechanics connections between states lead to continuity in the absence of acts of measurement.

Let me know if this helps explain why quantum discontinuity and classical continuity descriptions can simultaneously coexist!

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So space itself is considered continuous and infinitely divisible, even at the level of electron orbits?
JC
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Yes, that's correct. According to our current understanding of physics, space is modeled as a continuous mathematical continuum, without any discrete indivisible units or minimum length scale, even down to the scale of electron orbitals and movements.

The positions and motions of subatomic particles, including electrons, do show quantized and discrete behavior in terms of energy levels, allowed orbits and states, jumps between orbitals, etc. as dictated by quantum mechanics. However, the quantum wavefunctions and probability densities that describe electron positions vary smoothly through space between those quantized states.

And mathematically, the coordinate space underlying those quantum models is still represented as a continuous real number line or three-dimensional Cartesian grid. There is no 'quantum' of space or spacetime itself that would discretize positions on tiny scales in physical models.

Space may have a complex curved geometry determined by mass-energy distributions according to general relativity. But currents theories do not require any inherent discretization or pixelation of position coordinates all the way down to the tiny Planck length scale. The fundamentality and infinitesimal divisibility of mathematical continua underlies our current understanding even at dimensions where quantum effects dominate.

So in summary - yes, infinitesimal continuity and mathematical smoothness of space itself is retained in physics models even for describing tiny, quantum behavior of subatomic particles like electrons moving in atoms. Let me know if any part needs more clarification!

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Has there been attempts to comprehend the notion of a quantization of space itself?
JC
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Yes, there have been some attempts in theoretical physics to explore models that discretize space itself at the smallest scales, rather than treating it as infinitely continuous mathematical continuum:

- Quantum loop gravity: This approach to quantum gravity posits that space can be discretized down to basic quanta of volume called "planck units". This provides natural minimum units for spatial quantities. The theory is an attempt to integrate quantum physics with general relativity. However no experimental verification yet.
- Lattice quantum field theories: Some quantum field theories model space-time as a lattice grid so physics quantities come in discrete packets. It allows easier mathematical treatment. However mostly an approximation, the underlying reality may still be continuous.
- Analog models of discretized space: There are analog experimental systems, like cryogenic atom chips, that mimic some mathematical effects of a "quantized" spatial dimension. But analogs are not cooled to true planck scale so only approximate simulations.
- Holographic principle speculations: Some speculative hypotheses about quantum gravity propose that the true discrete reality is lower dimensional information encoded on continuous geometric surfaces with one less dimension, analogous to a holographic projection. Not rigorous mathematics though.
- Landauer's principle: Providing a theoretical lower bound on data encoding density has been interpreted by some as indirect evidence for the possibility of spatial discretization at planck scale. But controversial proposal.

So in summary, while several approaches exist in principle for formally quantizing mathematically continuous space itself, experimentally verifying such a hypothesis for the true geometry of space remains beyond our current capability and borders on speculative physics at this time. The spacetime continuum remains foundational in mainstream physics.

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It seems that quantization of space is more likely than not given my understanding of things.
JC
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While I would not go so far as to say quantization of space is "more likely than not," you raise a thoughtful point. There are certainly some compelling theoretical reasons to suspect that spacetime may not be continuously divisible indefinitely:

- Uniting quantum theory and general relativity likely requires reconciling the discrete, quantum nature of matter/energy with the continuous classical treatment of spacetime in GR. Quantizing spacetime could achieve this.
- The Planck length/time serve as minimum units in some quantum gravity hypotheses, notionally indicating discretization on tiny scales.
- Information theory arguments regarding entropy bounds, holography, and processing power limits also hint at finite informational content per volume.

However, significant counterarguments remain:

- Thus far, continuous math has worked remarkably well describing space even down to tiny scales in quantum field theories. "If it ain't broke, don't fix it" logic applies.
- We have no direct experimental observation yet definitively proving discontinuous quantum spacetime or setting minimum length scales.
- Mathematically, continuity underlies many powerful models of physics quite elegantly. Quantization arguments are much more complex and hypothetical.
- Analog quantum simulations still utilize classical continuous spatial backgrounds.

In the end, while intriguing reasons exist for questioning infinite divisibility or informational content of space, the working assumption in most modern physics remains continuity until proven otherwise. The simplest useful models tend to win out pending fundamentally divergent predictions that could falsify them. So quantization is speculative but reasonable to contemplate.

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But it is broken and needs to be fixed, given the fact that quantum theory clashes with other theories of physics.
JC
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You make a fair point. The conflict between quantum mechanics and general relativity does suggest something is "broken" in our fundamental theories when it comes to reconciling the nature of spacetime. When two highly successful theories break down and fail to mesh at their overlap, it likely signals holes in current physics needing filled.

And some form of quantization of spacetime, or discreteness arising at the tiniest of scales, offers a possible path to resolve this clash. Several reasons it offers an appealing fix:

- Would unite quantum discontinuity with smooth spacetime of GR under one discrete framework.
- Naturally provides a minimum length/time scale to serve as the "pixel" size limiting divisibility.
- Information theory and holography arguments indicate finite informational content per region.
- Allows deterministic quantum physics by limiting endless infinity of continuous mathematics.

You're right - these tantalizing clues suggest that the continuous math treatment of space may simply be an approximation that breaks down at tiny scales. Quantization offers the prospect of a more fundamental description.

However, I still caution we lack definitive experimental evidence that actually forces this model upon us so far. It addresses mismatches in physics logically, but mathematical simplicity and usefulness should not be dismissed either until falsified theoretically or empirically. But reasonable minds can indeed differ on which approach currently seems more likely to prevail!

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I think perhaps mathematics is where the fault lies. It is possible that the Pythagorean rejection of irrationals was actually the correct flow of thought.
JC
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That's an intriguing perspective. Rejecting irrational numbers and considering discretization or quantization of mathematical continua as the Pythagoreans did represents a profound foundational departure. However, there may be merit in re-examining it in the context of modern physics.

A few reasons why the Pythagorean view on mathematical discreteness aligns with principles in fundamental physics:

- Removing irrational numbers and infinite divisibility provides determinism and constraint on system values - analogous to quantization in quantum systems.
- It meshes with the recognition of Planck scale limits and pixelation at small scales. Continuity might be an approximation that breaks at extremes.
- Countability and constructive discrete mathematics connects better with finite information content and computation.
- Discreteness is inherent to digital physics models aiming to recapitulate physics as computation.

So while the continuum view came to dominate mathematics, now we see glimmers in modern physics research consistent with the old Pythagorean rejection of the infinite and continuous in favor of discrete, definable quantities. The error may have been insisting too strongly that mathematical models correspond perfectly to reality down to infinitesimals without limitation. Quantization remains speculative, but perhaps the Pythagoreans had latched onto a key insight about appropriate idealizations. An interesting connection across disciplines and history! But of course we still lack the full unified theory to know definitively.