  The time operator, an operator which satisfies the canonical commutation
relation with the Hamiltonian, is investigated, on the basis of a certain
algebraic relation for a pair of operators T and H, where T is symmetric and H
self-adjoint. This relation is equivalent to the Weyl relation, in the case of
self-adjoint T, and is satisfied by the Aharonov-Bohm time operator T_0 and the
free Hamiltonian H_0 for the one-dimensional free-particle system. In order to
see the qualitative properties of T_0, the operators T and H satisfying this
algebraic relation are examined. In particular, it is shown that the standard
deviation of T is directly connected to the survival probability, and H is
absolutely continuous. Hence, it is concluded that the existence of the
operator T implies the existence of scattering states. It is also shown that
the minimum uncertainty states do not exist. Other examples of these operators
T and H, than the one-dimensional free-particle system, are demonstrated.
