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

Jixin Chen

Published Aug 3, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.26434/chemrxiv-2022-xn4t8-v21

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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.
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.
Quantum mechanics presently has many unanswered questions, paradoxes, and even outright logical contradictions. To make progress in understanding quantum mechanics, we begin by proposing that relativity be set aside in favor of an absolute aetherial theory. Once that step is taken, we can understand quantum collapse as a description of real wave-packets collapsing in a faster-than-light way. By assuming that a partially observable reality exists, we can then extend our analysis of wave-packets into the subquantum, and the Heisenberg uncertainty principle then follows from the Fourier uncertainty principle coupled with the de Broglie relation. Further progress in understanding quantum mechanics is possible by modifying the de Broglie and Planck relations. Those modifications lead to matter-waves moving at the speed of light rather than superluminally as presently theorized, and they allow the results of matter-wave two-slit experiments to be understood from any reference frame. A modified time-dependent Schrödinger Equation results from our modifications, but the spatial time-independent Schrödinger equation is retained.
Quantum mechanics is our most successful physical theory. It predicts the behavior of atoms, molecules, and light to ten decimal places. No experiment has ever contradicted it. It also contains a crack at its foundation. The theory’s central equation—the Schrodinger equation—describes a smooth, continuous, perfectly predictable evolution of the quantum state. But whenever anyone performs a measurement, they get a single, definite result. Which result they get appears to be random. The transition from smooth evolution to random outcome is called the “collapse of the wavefunction,” and after nearly a century of analysis, there is no agreed-upon explanation for what causes it, when it occurs, or whether it occurs at all. This is the measurement problem. The measurement problem is not a minor puzzle at the margins of the theory. It is a failure to explain the most basic operation in science: looking at the result of an experiment. Around it cluster other puzzles. Light behaves like a wave in some experiments and like a particle in others—two contradictory descriptions. Two particles that have interacted remain correlated across arbitrary distances, as though communicating faster than light. The probabilities that govern measurement outcomes appear as a postulate—the Born rule—inserted by hand with no deeper justification. And the theory offers no unified picture of reality: quantum mechanics, general relativity, and the standard model of particle physics speak incompatible mathematical languages. A century of effort has produced many interpretations of quantum mechanics. Copenhagen. Many-worlds. De Broglie-Bohm. Objective collapse. Quantum Bayesianism. None has achieved consensus. The measurement problem remains. This document proposes that the source of these puzzles is not quantum mechanics itself. It is a single, invisible assumption made at the very beginning of the theory in 1900—an assumption so deeply embedded that nobody noticed they were making it. Change that assumption, and the puzzles do not get solved. They dissolve. They stop being puzzles. They become predictable consequences of geometry.