  An apparent paradox is resolved that concerns the existence of time operators
which have been derived for the quantum harmonic oscillator. There is an
apparent paradox because, although a time operator is canonically conjugate to
the Hamiltonian, it has been asserted that no operator exists that is
canonically conjugate to the Hamiltonian. In order to resolve the apparent
paradox, we work in a representation where the phase operator is diagonal. The
boundary condition on wave functions is such that they be periodic in the phase
variable, which is related to the (continuous) eigenvalue of the time operator.
Matrix elements of the commutator of the time operator with the Hamiltonian
involve the phase variable itself in addition to periodic functions of the
phase variable. The Hamiltonian is not hermitian when operating in space that
includes the phase variable itself. The apparent paradox is resolved when this
non-hermeticity is taken into account correctly in the evaluation of matrix
elements of the commutation relation.
