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

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¹ School of Physics and Information Science, Shaanxi University of Science and Technology, Xi’an 710021, China,

erd 2025 , 7(9), 198–204; https://doi.org/10.26689/erd.v7i9.12390

© 2025 by the Authors. Licensee Whioce Publishing, Singapore. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License ( https://creativecommons.org/licenses/by/4.0/ )

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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