Role of Fourier transform in quantum mechanics: applications and implications

The Fourier transform, as a fundamental mathematical tool, plays a pivotal role in quantum mechanics. Its significance extends to wave function analysis, solving the Schrdinger 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 Schrdinger 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