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Schrödinger equation for various quantum systems based on Heisenberg's uncertainty principle
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Abstract
This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.
Keywords
Fourier Transform, Heisenberg's Uncertainty Principle, Quantum Mechanics, Schrödinger Equation
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Cite this article
An,K. (2024). Schrödinger equation for various quantum systems based on Heisenberg's uncertainty principle. Theoretical and Natural Science,41,59-64.
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Volume title: Proceedings of the 2nd International Conference on Mathematical Physics and Computational Simulation
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