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arXiv:quant-ph/0604083

Wave equations for determining energy-level gaps of quantum systems

Zeqian Chen

Published Apr 12, 2006• 0 citations
arXivHTMLPDF

Abstract

An differential equation for wave functions is proposed, which is equivalent to Schrödinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schrödinger's wave equation, this equation is on `bipartite' wave functions. It is shown that those `bipartite' wave functions satisfy all the basic properties of Schrödinger's wave functions. Further, it is argued that `bipartite' wave functions can present a mathematical expression of wave-particle duality. This provides an alternative approach to the mathematical formalism of quantum mechanics.

DOI
10.48550/arxiv.quant-ph/0604083

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An differential equation for wave functions is proposed, which is equivalent to Schrödinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schrödinger's wave equation, this equation is on bipartite' wave functions. It is shown that those bipartite' wave functions satisfy all the basic properties of Schrödinger's wave functions. Further, it is argued that `bipartite' wave functions can present a mathematical expression of wave-particle duality. This provides an alternative approach to the mathematical formalism of quantum mechanics.
It is shown that `bipartite' wave functions can present a mathematical formalism of quantum theory for a single particle, in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This extension of Schrödinger's form establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. In particular, this formalism suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one. Our results shed considerable light on the basis of quantum mechanics, including quantum measurement.
It is shown that von Neumann-Landau equation for wave functions can present a mathematical formalism of motion of quantum mechanics. The wave functions of von Neumann-Landau equation for a single particle are `bipartite', in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This formalism establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. Furthermore, this extension of Schrödinger's form suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
It was Erwin Schrodinger who, in 1925, discovered an appropriate form of wave equation, making use of some deep formal analogies between optics and classical particle mechanics that had been evolved in the nineteenth century by W. R. Hamilton and others. There is all the accumulated evidence that the Schrodinger equations work; they provide the basis for a correct analysis of all kinds of molecular, atomic, and nuclear systems. The theoretical basis for the existence of states of discrete energies emerges more or less directly from the wave-particle duality; one of the most familiar features of classical wave behavior is that of standing waves. The quantum amplitudes for a particle in a bound state can be normalized. In order to make the statistical interpretation of the wave function more precise, we need to use the quantitative ideas of probability distributions.
Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.