Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Classical solution of the wave equation

М. Н. Сергеенко

Published Jan 1, 2001• 2 citations
OpenAlex

Abstract

The classical limit of wave quantum mechanics is analyzed. It is shown that the general requirements of continuity and finiteness to the solution ψ(x) = Aeiφ(x) + Be−iφ(x) , where φ(x) = 1 ¯h W(x) and W(x) is the reduced classical action of the physical system, result in the asymptote of the exact solution and general quantization condition for W(x), which yields the exact eigenvalues of the system. PACS number(s): 03.65.Sq, 03.65.Ge, 03.65.Yz 1. Introduction. One

Related papers

Classical Solution of the Wave Equation

M. N. Sergeenko

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.

arXiv:quant-ph/0010084•International Journal of Modern Physics A•Jul 10, 2003•3 citations

Quantization of the classical action and eigenvalue problem

М. Н. Сергеенко

The eigenvalue problem in quantum mechanics is reduced to quantization of the classical action of the physical system. State function of the system, $ψ_0(ϕ)$, 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 $ψ_0(ϕ)$ in the whole region.

arXiv:quant-ph/0211099•Nov 18, 2002•0 citations

Zeroth WKB Approximation in Quantum Mechanics

М. Н. Сергеенко

Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB$_0$ 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, $A_n\cos(k_nx+δ_n)$, and are the asymptote of the exact solution.

arXiv:quant-ph/0206179•Jun 25, 2002•3 citations

Semiclassical Mechanics

Joseph Bishop Keller

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.

SIAM Review•Dec 1, 1985•68 citations

On a Parallelism between Classical Mechanics and Quantum Mechanics

Yoshio Nishiyama

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.

Progress of Theoretical Physics•Aug 1, 1965•0 citations

15 Schrodinger Equation

Charles G Torre

An important feature of the wave equation is that its solutions q(r, t) are uniquely specified once the initial values q(r, 0) and (del)q(r, 0)/@t are specified. As was mentioned before, if we view the wave equation as describing a continuum limit of a network of coupled oscillators, then this result is very reasonable since one must specify the initial position and velocity of an oscillator to uniquely determine its motion. It is possible to write down other “equations of motion” that exhibit wave phenomena but which only require the initial values of the dynamical variable — not its time derivative — to specify a solution. This is physically appropriate in a number of situations, the most significant of which is in quantum mechanics where the wave equation is called the Schrodinger equation. This equation describes the time development of the observable attributes of a particle via the wave function (or probability amplitude) . In quantum mechanics, the complete specification of the initial conditions of the particle’s motion is embodied in the initial value of . The price paid for this change in the allowed initial data while asking for a linear wave equation is the introduction of complex numbers into the equation for the wave. Indeed, the values taken by are complex numbers. In what follows we shall explore some of the elementary features of the wave phenomena associated with the Schrodinger equation.

•Jan 1, 2014•0 citations
The classical limit of wave quantum mechanics is analyzed. It is shown that the general requirements of continuity and finiteness to the solution ψ(x) = Aeiφ(x) + Be−iφ(x) , where φ(x) = 1 ¯h W(x) and W(x) is the reduced classical action of the physical system, result in the asymptote of the exact solution and general quantization condition for W(x), which yields the exact eigenvalues of the system. PACS number(s): 03.65.Sq, 03.65.Ge, 03.65.Yz 1. Introduction. One
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.
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.
An important feature of the wave equation is that its solutions q(r, t) are uniquely specified once the initial values q(r, 0) and (del)q(r, 0)/@t are specified. As was mentioned before, if we view the wave equation as describing a continuum limit of a network of coupled oscillators, then this result is very reasonable since one must specify the initial position and velocity of an oscillator to uniquely determine its motion. It is possible to write down other “equations of motion” that exhibit wave phenomena but which only require the initial values of the dynamical variable — not its time derivative — to specify a solution. This is physically appropriate in a number of situations, the most significant of which is in quantum mechanics where the wave equation is called the Schrodinger equation. This equation describes the time development of the observable attributes of a particle via the wave function (or probability amplitude) . In quantum mechanics, the complete specification of the initial conditions of the particle’s motion is embodied in the initial value of . The price paid for this change in the allowed initial data while asking for a linear wave equation is the introduction of complex numbers into the equation for the wave. Indeed, the values taken by are complex numbers. In what follows we shall explore some of the elementary features of the wave phenomena associated with the Schrodinger equation.