Schrödinger-like Equations and Wave Mechanics: Bridging Quantum and Classical Realms

Abstract

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.