Quantum time transition states

Abstract

We show that for a particle spatially confined in a potential-free segment of the real line, there exists a self-adjoint time operator, known as the Characteristic Time Operator (CTO), which forms a dense canonical pair with the Hamiltonian corresponding to the system. The CTO can be interpreted in terms of the internal dynamics of the system, as follows: it is an operator whose eigenfunctions corrsponding to small-magnitude eigenvalues evolve in such a way that they make transitions to other CTO eigenfunctions which also correspond to small-magnitude eigenvalues. We compare the dynamical behavior of the CTO with the dynamical behavior of the self-adjoint Confined Quantum Time of Arrival (CTOA) operator corresponding to the system, which forms a closed canonical pair with the Hamiltonian.