Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator

Abstract

The quantum measurement axiom dictates that physical observables and in particular the Hamiltonian must be diagonalizable and have a real spectrum. For a time-independent Hamiltonian (with a discrete spectrum) these conditions ensure the existence of a positive-definite inner product that renders the Hamiltonian self-adjoint. Unlike for a time-independent Hamiltonian, this does not imply the unitarity of the Schroedinger time-evolution for a general time-dependent Hamiltonian. We give an additional necessary and sufficient condition for the unitarity of time-evolution. In particular, we obtain the general form of a two-level Hamiltonian that fulfils this condition. We show that this condition is geometrical in nature and that it implies the reality of the adiabatic geometric phases. We also address the problem of the uniqueness of the metric operator.