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arXiv:quant-ph/0010084 · International Journal of Modern Physics A

Classical Solution of the Wave Equation

M. N. Sergeenko

Published Jul 10, 2003• 3 citations
arXivHTMLPDF

Abstract

The classical limit of wave quantum mechanics is analyzed. It is shown that the basic requirements of continuity and finiteness to the solution of the form ψ(x) = Aei ϕ (x) + Be-i ϕ (x), where [Formula: see text] and W(x) is the reduced classical action of the physical system, give the asymptote of the wave equation and general quantization condition for the action W(x), which yields the exact eigenvalues of the system.

DOI
10.1142/s0217751x03015805

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The classical limit of wave quantum mechanics is analyzed. It is shown that the basic requirements of continuity and finiteness to the solution of the form ψ(x) = Aei ϕ (x) + Be-i ϕ (x), where [Formula: see text] and W(x) is the reduced classical action of the physical system, give the asymptote of the wave equation and general quantization condition for the action W(x), which yields the exact eigenvalues of the system.
The eigenvalue problem in quantum mechanics is reduced to quantization of the classical action of the physical system. State function of the system, ψ0(ϕ)ψ_0(ϕ)ψ0​(ϕ), is written in the form of superposition of two plane waves in the phase space. Quantization condition is derived from the basic requirements of continuity and finiteness for ψ0(ϕ)ψ_0(ϕ)ψ0​(ϕ) in the whole region.
Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB0_00​ method. Solution in the whole region is built with the help of simple connection formulas we derive from basic requirements of continuity and finiteness for the wave function in quantum mechanics. We show that, for conservative quantum systems, not only total energy, but also momentum is the constant of motion. We derive the quantization conditions for two and more turning point problems. Exact energy eigenvalues for solvable and some ``insoluble'' potentials are obtained. The eigenfunctions have the form of a standing wave, Ancos⁡(knx+δn)A_n\cos(k_nx+δ_n)An​cos(kn​x+δn​), and are the asymptote of the exact solution.
Abstract In many texts, the transition from classical mechanics to quantum mechanics is achieved by substituting the action for the phase angle. The paper presents a different approach to show some connections between classical and quantum mechanics for a single particle for an audience at graduate and postgraduate levels. Firstly, it is shown that a wave equation of action can be derived under the free particle condition and the Legendre transform. The wave-like solutions of the action, Hamiltonian and momentum of the free particle are presented. Using the discrete approximation, the equation of motion of a single particle, in scalar potential field, is obtained in a similar form to Schrödinger’s equation. The rest of the paper discusses the propagation, superposition of the wave-like dynamic variables and their connections to quantum mechanics. The superposition of the variables of a particle is generally distinct from the superposition of classical waves (e.g. acoustics). The quantum superposition provides a self-consistent interpretation of the wave-like solutions of the variables. Connections between the classical and quantum relations for corresponding variables are observed from the one-to-one comparisons.
Classical mechanics and the quantum conditions of Planck, Bohr, Sommerfeld, Wilson and Einstein are presented. The virtues and defects of this “old quantum theory” are pointed out. Its replacement by quantum mechanics is described, leading to the Schrödinger equation for the wave function and the corresponding energy eigenvalues. For separable systems, the reduction of this equation to ordinary differential equations and their asymptotic solution by the WKB method are described, as well as the resulting corrected quantum conditions with integer or half-integer quantum numbers. For nonseparable systems, the analogous asymptotic solution constructed by the author is described, together with the corrected quantum conditions to which it leads. Examples of the use of these conditions in the solution of eigenvalue problems are presented. It is explained that difficulties arise in using this method when the classical motion is stochastic or chaotic. Suggestions for overcoming these difficulties are mentioned.
Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound system in a rather general form by introducing reasonably “quantum-mechanical momentum” from the Schrödinger equation. The following are proved: The “quantum-mechanical momentum” which is nearly equal to the corresponding classical momentum in “classical region” can be defined in the whole range of validity of the coordinate for the bound systems. In the same way as in classical mechanics, in quantum mechanics one can construct an associated wave accompanying a motion of a quantum particle and obeying the Schrödinger equation. The uniqueness condition on the associated wave throughout an orbit leads to the boundary condition on the wave function which is composed of the associated wave and to “quantum-mechanical quantization condition”, equivalently. In addition, all these quantum-mechanical quantities are approximately equal to the corresponding classical ones in the “classical region” of the coordinate. In particular it is demonstrated that even for a bound system of a singular (central) potential exists the ground state with the lowest energy eigenvalue. It is proved in one dimension that to any physical system obeying the Schrödinger equation always exists the corresponding classical system.