Quantum Mechanical Operator of Time

TL;DR

This paper introduces a self-adjoint operator for time in non-relativistic quantum mechanics, exploring its implications for the Liouville and von Neumann equations with quantized time, aiming to improve understanding of exact time measurement.

Contribution

It presents a novel self-adjoint time operator and reformulates the Liouville and von Neumann equations incorporating quantized time, which is a new approach in quantum mechanics.

Findings

Derived a self-adjoint time operator in quantum mechanics.

Reexpressed the Liouville equation with a total Hamiltonian including conjugate time.

Discussed the von Neumann equation with quantized time for precise time measurement.

Abstract

The self adjoint operator of time in non-relativistic quantum mechanics is found within the approach where the ordinary Hamiltonian is not taken to be conjugate to time. The operator version of the reexpressed Liouville equation with the total Hamiltonian, consisting of the part that is a conventional function of coordinate and momentum and the part that is conjugate to time, is considered. The von Neumann equation with quantized time is found and discussed from the point of view of exact time measurement.