Existence and stability results for time-fractional Schrödinger equations Related to the harmonic oscillator

Abstract The implications of the Schrödinger equation is profound, revealing the wave-particle duality of matter, the concept of superposition, and the probabilistic nature of quantum measurements. Its solutions provide critical insights into the behavior of atoms, molecules, and subatomic particles, forming the foundation for much of modern physics and technology. This article is concerned with the existence and uniqueness of solutions for Schrödinger equation that involve fractional differential equations using the Caputo method with initial conditions, as well as exploring the Hyers-Ulam stability of the Schrödinger equation was investigated when the relevant system has a potential well of finite depth using a fixed-point approach. Finally, we present the graphical representation for the Schrödinger equation related to the harmonic oscillator.