A stationary formulation of time-dependent problems in quantum mechanics

Abstract

A unifying framework is presented which treats time-dependent problems in nonrelativistic quantum mechanics on equal footing with stationary ones. This is accomplished by elevating time t to the role of a dynamical variable and considering evolution in an extended space with respect to a progress variable τ. Among the new objects in the extended space are the time operator as well as operators, such as energy, that are not diagonal in time. The latter bear, e.g., on the question of ‘‘quantum chaos.’’ Only τ-stationary states in the extended space are necessary to recover all of the Schrödinger time evolution. In particular, time-dependent constants of the motion (and these include time-evolved density matrices) elevate to stationary constants of the τ motion. Time-dependent problems, which may also involve time-dependent Hamiltonians, can then be solved by stationary-state methods. Special attention is given to applications based on time-dependent constants of the motion and to maximum-entropy states subject to time-dependent constraints. Two examples (rank-one perturbation, damped harmonic oscillator) illustrate some of these ideas. Scattering theory in the extended space is discussed in two appendices. It furnishes a picture of Hamiltonian τ evolution which promises to embrace also irreversible dynamics.