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From Particle-in-a-Box Thought Experiment to a Complete Quantum Theory?

Jixin Chen

Published Oct 31, 2023• 1 citation

Abstract

While instructing an undergraduate physical chemistry course on quantum mechanics, I encountered difficulties in elucidating the concept of the particle-in-a-box thought experiment, initially formulated by Erwin Schrödinger. In particular, the node distribution of the wavefunction. Born's Copenhagen interpretation assigns the modulus of the wavefunction as the probability density distribution of particles. The developers of quantum mechanics, including Schrödinger, de Broglie, and Einstein, have openly expressed reservations about this interpretation. Schrödinger's original conception of the wavefunction is intrinsically linked to the particle's momentum rather than the statistical probabilities of particle locations. Because momentum can be encoded in different forms, it could be useful to create wavefunctions using different encoding equations, which is the major attempt of this manuscript. In addition to quantizing energy levels properly for several example systems, the proposed new equations have yielded a striking and alternative interpretation of Schrödinger's wavefunction that aligns well with classical mechanics. This proposed modification enhanced the simplicity of math and strengthened Schrödinger’s original interpretation. These equations could be baby steps to a brand-new quantum calculation algorithm that is simpler and faster than the established ones.

DOI: 10.26434/chemrxiv-2022-xn4t8-v23

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It is an unavoidable question raised by Albert Einstein if quantum theory is complete. The wavefunctions of elements are now constructed from fitting the experimental data which has made quantum chemistry a black box of mathematical model. It is our best interest to come up with a simpler model to fit the data regardless of our wish to understand reality in a more intuitive and accurate way than classical quantum mechanics. In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform according to the 1st law of thermodynamics. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The radial wavefunctions for the hydrogen atom and an ideal harmonic oscillator are then derived. Can this assumption on energy distribution and the pre-mature math in this manuscript lead to a complete quantum theory? Can the idea of energy conservation unify special relativity and quantum?

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It is an unavoidable question raised by Albert Einstein if quantum theory is complete. The wavefunctions of elements are now constructed from fitting the experimental data which has made quantum chemistry empirical for almost all elements and molecules. It is our best interest to come up with a simpler model to fit the data regardless of our wish to understand reality in a more intuitive and accurate way than classical quantum mechanics. In this manuscript, I proposed an equation that quantizes the energies orders of magnitude faster than classical quantum mechanics. I assumed that the energy density distributions along space and time in a quantum system are uniform according to the 1st law of thermodynamics. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with an equation significantly different than the classical Schrödinger equation. This new equation is inspired by the energy conservation method used in solving classical mechanics problems such as the pendulum problem. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The radial wavefunctions for the hydrogen atom and harmonic oscillator are derived. We can see that the new equation is much simpler than Schrödinger equation in obtaining consistent quantized rotational and vibrational energy levels. Thus, it has the potential to be widely used to free up computational pressure in modern quantum chemistry calculations.

Einstein: God does not play dice and nothing travels faster than the speed of light. Schrödinger: My cat cannot be both dead and alive. Dirac: The existing quantum theory should not be considered to be the final solution. Feynman: How does it work? de Broglie: My matter wave has an imaginary part. Me: Great! This could be the missing piece the others have been looking for. In this report, I present an innovative modification to the Schrödinger equation, which yields wavefunctions capable of accurately describing quantum phenomena. Concurrently, this modification introduces an alternative interpretation to the widely recognized Copenhagen interpretation within the realm of quantum theory.

Einstein: God does not play dice and nothing travels faster than the speed of light. Schrödinger: My cat cannot be both dead and alive. Dirac: The existing quantum theory should not be considered to be the final solution. Feynman: How does it work? de Broglie: My matter wave has an imaginary part. Me: Great! This could be the missing piece the others have been looking for. In this report, I present an innovative modification to the Schrödinger equation, which yields wavefunctions capable of accurately describing quantum phenomena. Concurrently, this modification introduces an alternative interpretation to the widely recognized Copenhagen interpretation within the realm of quantum theory.

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