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Mianyang Shifan Xueyuan xuebao

Discussion on the Transition from Schrdinger Equation to Classical Mechanics

Xuedong Li

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

Schrdinger Equation in quantum mechanics is as important as Newton Equation in classical me- chanics.From the point of Schr(?)dinger Equation,two different methods are employed to discuss the problem of the transition from quantum mechanics to classical mechanics.The result shows that under the condition of Planck con- stanth→O quantum mechanics goes hack to classical mechanics and the motion of micro particles follows Newton's second law from the angle of average value,which means quantum effect is only a kind of fluctuation around aver- age value,namely quantum fluctuation.

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Schrdinger Equation in quantum mechanics is as important as Newton Equation in classical me- chanics.From the point of Schr(?)dinger Equation,two different methods are employed to discuss the problem of the transition from quantum mechanics to classical mechanics.The result shows that under the condition of Planck con- stanth→O quantum mechanics goes hack to classical mechanics and the motion of micro particles follows Newton's second law from the angle of average value,which means quantum effect is only a kind of fluctuation around aver- age value,namely quantum fluctuation.
This study examines integrating the Schrödinger equation with classical mechanics using a virtual axis-to-dimensional expansion. One-dimensional material fluctuations are viewed in a two-dimensional plane, explaining the random nature of these fluctuations and their spatial and temporal trajectories. A quantum-consistent force field is proposed, with its strength determined by the Planck constant and inversely proportional to the distance from the stationary point. Newton's second law is applied to establish a second-order linear differential equation for material fluctuations, from which the standard one-dimensional Schrödinger equation is derived, showing their equivalence. The study extends the three-dimensional Schrödinger equation to include external forces and explains quantum phenomena like energy levels and transitions through particle trajectory changes. This approach connects classical mechanics and quantum mechanics, offering a concise and intuitive formulation with clear physical significance.
Quantum mechanics and classical mechanics are closely related to each other.The Hamilton-Jacobi Equation of classical mechanics plays an important role in the extension of Schrdinger equation,and it is also a way to extend Schrdinger equation in the teaching of quantum mechanics.Besides,the Hamilton-Jacobi Equation under the theory of relativity can extend Klein-Gordon equation of quantum mechanics of special relativity.Considering and analyzing the differences betweeen Klein-Gordon equation and Dirac Equation further,this paper linearized the Hamilton-Jacobi Equation under the theory of relativity,and extended Dirac Equation which was more important than quantum mechanics of special relativity.It made the route by which Hamilton-Jacobi Equation,as the classical mechanics,goes to quantum mechanics more extensive,and this study further explored the corresponding relationship between classical mechanics and quantum mechanics.
Schordinger's equation is the basical equation of quantum mechanics,which is equivalent to Newton's motion equation in classical physics. This paper compares the differences between quantum mechanics and classical physics by means of discussing the description of motion state and the expression of mechanic quantity of microcosmic particles in Schordinger's equation. It can be seen in this paper that the basical principle of quantum mechanics is the statistical one, while the basical principle of classical physics is the determinism and the strict causality. But under the h→0 limit in Planck constant, quantum mechanics will go beyond normal limit to classical physics.
Based on the theory of Schrdinger equation for the free particle,the solution of that equation was discussed in detail. It is found that the equation only exist an insignificant solution. And the mediocre solution can be also obtained from the one-dimensional infinitely deep potential well in a potential well width tending to infinite and from one-dimensional square potential barrier in the barrier height tending to zero. Although the solution is insignificant,but it describes some ideal that is consistent with that of classical mechanics. It explains the connection between the classical mechanics and quantum mechanics. The mediocre solutions show that 1) For infinite free particles,it is followed by Newton' s first motion law,2) For infinite free particle,it does not exist matter wave characteristics. Only if there is some potential( i. e. interactions),the Wave-particle duality be embodied.
The following equation pair shows the close connection between quantum and classical mechanics. F = m × a = ×( ) ( )average F m average a The first equation is the basis of classical mechanics. It summarizes Newton’s laws, where F is the force, m is the mass, and a is the acceleration. The second equation is the quantum-mechanical generalization. It looks almost the same except for the word average. Averaging is necessary because particles in quantum mechanics are associated with waves, and one cannot precisely determine the position or motion of a wave. Only the averages are known.