Matrix representation of the time operator

In quantum mechanics the time operator $\mathrm{\Theta }$ satisfies the commutation relation $[\mathrm{\Theta },H]=i$, 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 $\mathrm{\Theta }$ by $\Theta+ \Theta_{\rm hom}$,where$\Theta_{\rm hom}$satisfiesthehomogeneouscondition$[\Theta_{\rm hom},H]=0$\mathrm{.}Tostudythisnonuniquenessthematrixelementsof$\Theta$ 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 $\mathrm{\Theta }$ simplify dramatically. However, it is still not clear whether there is an optimally simple representation of the time operator.