Matrix representation of the time operator

Abstract

In quantum mechanics the time operator $Θ$ satisfies the commutation relation $[Θ,H]=i$, 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 $Θ$ by $Θ+ Θ_{\rm hom}$, where $Θ_{\rm hom}$ satisfies the homogeneous condition $[Θ_{\rm hom},H]=0$. To study this nonuniqueness the matrix elements of $Θ$ 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 $Θ$ simplify dramatically. However, it is still not clear whether there is an optimally simple representation of the time operator.