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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Closed-form solutions for the modified potential

Edward R. Floyd

Published Nov 15, 1986• 48 citations
Publisher

Abstract

Heretofore, the modified potential had to be described by approximation. Now, closed-form solutions may be developed for the modified potential. Closed-form solutions may now prove the hypotheses, which heretofore could only be shown to be plausible, that the quantum-action-variable quantization is precisely consistent with quantization of wave mechanics and that the Schr\"odinger wave function has microstates. An alternate representation of the set of nonlocal hidden variables for determining quantum continuous motion may be developed in terms of the closed-form solutions. The relationship between this alternate representation for the set of nonlocal hidden variables and Ermakov's invariant may now be developed.

DOI
10.1103/physrevd.34.3246

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Heretofore, the modified potential had to be described by approximation. Now, closed-form solutions may be developed for the modified potential. Closed-form solutions may now prove the hypotheses, which heretofore could only be shown to be plausible, that the quantum-action-variable quantization is precisely consistent with quantization of wave mechanics and that the Schr"odinger wave function has microstates. An alternate representation of the set of nonlocal hidden variables for determining quantum continuous motion may be developed in terms of the closed-form solutions. The relationship between this alternate representation for the set of nonlocal hidden variables and Ermakov's invariant may now be developed.
The relationship between the modified potential and Bohm's quantum-mechanical potential is developed. For a given bound-state energy eigenvalue of the Schr\"odinger equation, the modified potential, which is not unique, describes various possible continuous motions. It is shown that all of the various continuous motions of the same energy eigenvalue do generate the same corresponding Schr\"odinger wave function. The parameters which specify these different continuous motions are identified as hidden variables. The equations of motion for a particle with the modified potential differ from those of Bohm.
The form of Schrödinger's equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set 𝔖 of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain 𝔖 is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.