Uniform operator: Aligning fractional time quantum mechanics with basic physical principles

We propose a fractional time corresponding operator to elucidate the relationship between fractional time and total energy. We then use the operator to derive the fractional time Schrödinger equation (UFTSE), which exhibits spectral equivalence with a comb model for [Formula: see text] (Eq. ( 18 )). Based on the UFTSE framework, we demonstrate four basic principles of fractional quantum mechanics: (a) the unitarity of the time-evolution operator, (b) the existence of stationary energy spectra, (c) the probability conservation law and (d) the Heisenberg time–energy uncertainty principle. For typical applications, we applied a uniform equation to study three typical particles: free particle, particle in an infinite potential well and particle in a rectangular barrier. This uniform equation brings about some typical physical phenomena, including (1) nonsymmetric basic wave functions, (2) a continuous energy spectrum for a free particle and an infinite potential well and (3) tunneling through a rectangular barrier. These phenomena, which have not been observed in previous studies or standard quantum mechanics, will likely attract further applications in various scientific fields.