The Principles of Quantum Mechanics. I. Classical Waves and the Schrödinger Equation

Reinhold Fürth

Oleg A. Chevtchenko

This study delves into the realm of Schrödinger equations and wave mechanics, aiming to see if they could provide a foundation comparable in generality to that formulated in the preceding chapter, and to bridge the gap between classical and quantum physics. It begins by deriving a family of Schrödinger-like equations. The study introduces an intuitive approach for deriving an entire spectrum of Schrödinger-like equations, encompassing classical and novel variants, through the utilization of the complex wave function and its conjugate, along with the assumptionsA01, A08. Examining the concepts of the complex wave function \(\psi\) and the multi-vector wave function \(\Psi\) reveals their equal generality, suggesting the applicability of Schrödinger-like equations in the context of multi-vectors, including both \(\psi\) and the phase \(\Phi\) . To explore the correspondence principle between quantum and classical mechanics, a well-studied example is dissected in Section 2.4, necessitating the development of a suitable wave function (2.50). The use of the real function [1, (1.50)] to form the complex test function (2.50) is a logical step, and its substitution into the Schrödinger equations aims to yield an identity. The substitution of (2.50) reveals a discrepancy between the left and right sides of the Schrödinger equations (2.25, 2.26), highlighting their inadequacy in describing the classical mechanics example accurately. This observation underscores the limitations of classical Schrödinger equations and their departure from the correspondence principle. Consequently, a more comprehensive set of Schrödinger-like equations (2.58-2.68) is derived, rendering the previously considered equations (2.25, 2.26) irrelevant. Substituting the test function (2.50) into equations (2.58-2.68) demonstrates their soundness and suitability for the example, establishing the correspondence principle between classical and quantum mechanics. Our analysis indicates that the classical set [1, (1.4-1.6)] exhibits a higher degree of generality than the quantum one (2.58-2.68), thereby deferring the definitive validation of this assertion to forthcoming research endeavours. These findings underscore the potential significance of our framework in bridging the gap between quantum and classical mechanics, promising further exploration and validation in future studies.

Kim Fook Lee

Abstract Just as it is impossible to derive Newton ‘s equations of motion in classical mechanics or Maxwell ‘s equations for electricity and magnetism (EM) from first principles, neither can we demonstrate the validity of the Schrödinger equation approach (or any other equivalent one) to quantum mechanics a priori. We can, however, make use of early quantum ideas and the connection between the classical wave equation and the photon concept to help in understanding its structure.

John P. Lowe

Chengyuan ZHANG, Yong Ding, Guangzhi XU +2 more

The Schrödinger equation is the fundamental equation of quantum mechanics, primarily discussing the evolution of quantum states over time and solving the wave function under specific conditions. Its position is equivalent to that of Newton's equation in classical mechanics. Based on the literature “Teaching Attempts to Introduce Wave-Particle Duality in Quantum Mechanics,” this paper further establishes the Schrödinger equation through the Maupertuis principle. By comparing several typical teaching methods for establishing the Schrödinger equation, the characteristics of the method proposed in this paper are further highlighted, which helps students to integrate new and old knowledge organically, more easily accept and understand the basic equations of quantum mechanics, enhance students' ability to transfer knowledge, and stimulate students' creative thinking.