Quantization method and Schrödinger equation of fractional time and their weak effects on Hamiltonian: Phase transitions of energy and wave functions

Abstract

Fractional time quantum mechanics (FTQM) is a method of describing the time evolution of quantum dynamics based on fractional derivatives. For any potential, we obtain the modeling method to describe quantum systems with fractional time consistent with fundamental quantum physics laws, which makes fractional time effects naturally enter quantum mechanics. The method is only using the start states, not based on usually directly replacing the integer derivatives by fractional ones or parallel introductions of standard models. In the process, we solve three open problems perplexing the studies on FTQM: What is the essential quantization method? How does one represent the fractional time Hamiltonian while retaining physical significance? How does one avoid the violations of current models of FTQM for many fundamental quantum physics laws? Then, a FTQM framework is built by amalgamating two quantization methods under a unified foundation of fractional time. The framework contains the quantization method, the Hamiltonian, the Hamilton operator, the Schrödinger equation, the energy correspondence relation, the Bohr correspondence principle and the time-energy uncertainty relation. And the effects of fractional time are revealed: containing historical information of particle’s motions; representing weak actions of Hamilton operator. An example is provided, and analytic expressions of the energy and wave functions are obtained. These account for the distinct nonlinear phenomena: the phase transition of energy and wave functions, which can not be revealed in the previous methods. The phase transitions cause some classical physical effects and phenomena: (1) energy gaps filled by energy levels; (2) increase in particle orbits; (3) a famous bound states in continuum (BICs) firstly found in FTQM; (4) a new explanation of the discrete energy levels from the perspective of energy level filling.