A Study on the Teaching of Mathematical Methods for Physics based on the Laws of Scientific Cognition: The Introduction of Complex Numbers as an Example

Yong Niu; Ying Wang; Linhao Wang

doi:10.26689/erd.v7i9.12390

Yong Niu School of Physics and Information Science, Shaanxi University of Science and Technology, Xi’an 710021, China
Ying Wang School of Physics and Information Science, Shaanxi University of Science and Technology, Xi’an 710021, China
Linhao Wang School of Physics and Information Science, Shaanxi University of Science and Technology, Xi’an 710021, China

DOI: https://doi.org/10.26689/erd.v7i9.12390

Keywords: Complex numbers, Visualization of complex numbers, Schrödinger equation, Scientific cognition principles

Abstract

The development of innovative talent should be guided by the principles of scientific cognition. Currently, there is a significant gap in students’ understanding of the origin, visualization, and necessity of complex numbers, particularly in their application to physics. This paper examines the historical development of complex numbers, tracing contributions from Ferro, Cardano, and Bombelli to Gauss’s systematic geometric representation. It emphasizes the necessity and geometric significance of introducing complex numbers. Additionally, the paper explores the critical role of complex numbers in physics, especially in quantum mechanics. Using the Schrödinger equation as a case study, we demonstrate that the introduction of complex numbers not only ensures the existence of solutions but also provides a natural framework for describing the phase evolution and probability amplitudes of wave functions for microscopic particles. Through a thorough analysis of the motivations for the introduction and application of complex numbers, this paper aims to enhance students’ understanding of their mathematical nature, geometric visualization, and physical significance, ultimately contributing to the development of innovative talent.

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