Time evolution for quantum systems with a dynamical Hilbert space

Abstract

We investigate the time evolution for quantum systems with a dynamical Hilbert space within the framework of the pseudo-Hermitian representation of quantum mechanics. Each representation of quantum mechanics is characterized by a Hilbert space, a Hamiltonian, and a set of observables. The representation-transformation law of the time-evolution operator is derived from its formal solution in the dynamical Hilbert space. It ensures the unitarity of the dynamics and the representation independence of the transition amplitudes. In addition, we show that the equation of time evolution in the position space is independent of the representation, regardless of whether the Hilbert space is stationary or dynamical. Furthermore, we demonstrate the representation independence of the position wave function itself. As a concrete example of the representation independence of the quantum mechanics, we derive the reciprocity theorem in the dynamical Hilbert space. The material in the present paper makes a topic which can be covered in a graduate course on quantum mechanics.