Operator of Time and Generalized Schrödinger Equation

Slobodan Prvanović

doi:10.1155/2018/6290982

Abstract

The equation describing the change of the state of the quantum system with respect to energy is introduced within the framework of the self-adjoint operator of time in nonrelativistic quantum mechanics. In this proposal, the operator of time appears to be the generator of the change of the energy, while the operator of energy that is conjugate to the operator of time generates the time evolution. Two examples, one with discrete time and the other with continuous one, are given and the generalization of Schrödinger equation is proposed.

Highlights

The most important equation in quantum mechanics is the Schrodinger equation [1]
It is seen as the equation that, for a particular choice of the Hamiltonian, determines how the state of the quantum mechanical system changes with respect to time
The proposed formalism, we believe, can be applied in the discussion of the situations when a quantum system is energy-driven by an external field, so its energy continuously changes, or in the cases with the instantaneous change of the energy in an interaction quench

Summary

Introduction

The most important equation in quantum mechanics is the Schrodinger equation [1]. The Hamiltonian is a function of coordinate and momentum, H(q, p), but there are situations in which one is interested in time dependent Hamiltonians as well [2,3,4,5,6,7,8] In these cases, one can find how the energy of the system changes in time, which is the consequence of the influence of the environment on the system under consideration. In order to tackle this problem, one needs an equation in which, so to say, one differentiates with respect to the energy, not with respect to the time as in the Schrodinger equation This is what we are going to discuss in the present article, and this will be done by using the formalism of the operator of time. Our approach is similar to the one in [16] and the references therein and [17]

Operators of Time and Energy and Schrödinger Equations

Concluding Remarks