Matrix representation of the time operator

TL;DR

This paper investigates the matrix representation of the quantum time operator for the harmonic oscillator, addressing its nonuniqueness and simplifying its matrix elements using zeta-summation techniques.

Contribution

It provides a detailed calculation of the time operator's matrix elements, including homogeneous contributions, and explores their simplification.

Findings

Matrix elements simplified with homogeneous contributions

Use of zeta-summation techniques for divergent series

Nonuniqueness of the time operator representation

Abstract

In quantum mechanics the time operator $\Theta$ satisfies the commutation relation $[\Theta,H]=i$, and thus it may be thought of as being canonically conjugate to the Hamiltonian $H$. The time operator associated with a given Hamiltonian $H$ is not unique because one can replace $\Theta$ by $\Theta+ \Theta_{\rm hom}$, where $\Theta_{\rm hom}$ satisfies the homogeneous condition $[\Theta_{\rm hom},H]=0$. To study this nonuniqueness the matrix elements of $\Theta$ for the harmonic-oscillator Hamiltonian are calculated in the eigenstate basis. This calculation requires the summation of divergent series, and the summation is accomplished by using zeta-summation techniques. It is shown that by including appropriate homogeneous contributions, the matrix elements of $\Theta$ simplify dramatically. However, it is still not clear whether there is an optimally simple representation of the time operator.