A generalized Weyl relation approach to the time operator and its connection to the survival probability

Abstract

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.

16 pages, REVTeX. Accepted for publication in the Journal of Mathematical Physics

References in corpus (2)

• Free motion time-of-arrival operator and probability distribution
• Time-of-arrival distribution for arbitrary potentials and Wigner's time-energy uncertainty relation

Cited by in corpus (8)

• The various power decays of the survival probability at long times for free quantum particle
• Strong time operators associated with generalized Hamiltonians
• Ultra-Weak Time Operators of Schroedinger Operators
• Weak commutation relations of unbounded operators: nonlinear extensions
• Matrix representation of the time operator
• A Hardy's Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra
• Time delay and Calabi invariant in classical scattering theory
• Late-time universal distribution functions of observables in one-dimensional many-body quantum systems