A novel kind of equations linking the quantum dynamics and the classical wave motions based on the catastrophe theory

¹ School of Mechanical Engineering, Xi'an Jiaotong University & State Key Laboratory for Strength and Vibration of Mechanical Structures - Xi'an 710049, China
² College of Information Science and Engineering, Zhejiang University - Hangzhou 310027, China

^(a) ejhwu@mail.xjtu.edu.cn (corresponding author)
^(b) zhlforever@stu.xjtu.edu.cn

Received: 2 June 2021
Accepted: 29 September 2021

Abstract

Considering that the catastrophe theory could describe quantitatively any phase transition process, we adopt the folding and cusp catastrophe types as the potential functions in the Schrödinger equation to attempt to link the quantum dynamics and the classical wave motions. Thus, through the dimensionless analysis a novel kind of partial differential equations is derived out. When the scaling parameter of the novel equation is equal to the Planck's constant, this equation becomes a detailed time-independent Schrödinger equation, from which Bohr correspondence principle can be found. On the other hand, when the scaling parameter tends to zero, this equation could degenerate to the classical Helmholtz equation. Therefore, this novel kind of equations could describe quantitatively the variation process of the wave functions from the macroscopic level to the quantum size.