Operator of Time and Generalized Schroedinger Equation

TL;DR

This paper introduces a generalized Schrödinger equation based on a self-adjoint time operator in non-relativistic quantum mechanics, proposing a new framework where time and energy operators generate each other's evolution.

Contribution

It presents a novel formulation of quantum dynamics using a self-adjoint time operator, extending the traditional Schrödinger equation to include generalized time-energy relations.

Findings

Time operator acts as generator of energy change.

Energy operator generates time evolution.

Generalized Schrödinger equation is proposed.

Abstract

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