Starting from the definition of Fourier transform, uncertainty principle in quantum mechanics is educed by the short term Fourier transform for the time variable signal. And the necessity of the relation between them is explained in this article.
The Fourier transform, as a fundamental mathematical tool, plays a pivotal role in quantum mechanics. Its significance extends to wave function analysis, solving the Schrödinger equation, and elucidating the relationship between position and momentum. In this review article, the primary objective is to summarize the diverse applications of the Fourier transform in quantum mechanics. This paper will delve into key applications, highlighting the uncertainty principle, the Planck-Einstein relation, and the Fourier transform solution of the Schrödinger equation. These theorems and relationships not only facilitate the mathematical manipulation of quantum states but also lay the theoretical foundation of quantum mechanics. By exploring these critical aspects, the author aims to provide a comprehensive understanding of how the Fourier transform underpins the core principles of quantum theory, offering valuable insights into the wave particle duality and the probabilistic nature of quantum systems. This discussion will emphasize the essential nature of the Fourier transform in both theoretical development and practical problem-solving within quantum mechanics
We obtain both a global and local uncertainty principle for the short-time Fourier transform. Explicit expressions are obtained for the uncertainty product in terms of the uncertainty product of the signal and window. Although the uncertainty product of the spectrogram is not the sum of the uncertainty products of the window and signal, nonetheless, we show that the global uncertainty product has to be greater than one. We also derive local uncertainty relations which indicate how the local spread in time and frequency are related at a particlar time–frequency point. In addition, we obtain general results for the time, scale, and frequency moments of the scalogram in terms of the moments of the signal and mother wavelet. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.