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

TL;DR

This paper explores conditions under which time-dependent pseudo-Hermitian Hamiltonians define a unitary quantum system, providing a general form for two-level systems and discussing the uniqueness of the metric operator.

Contribution

It establishes a necessary and sufficient geometric condition for unitarity in time-dependent pseudo-Hermitian quantum systems and derives the general form for two-level Hamiltonians.

Findings

Identifies a geometric condition for unitarity of time-dependent Hamiltonians.

Derives the general form of two-level Hamiltonians satisfying this condition.

Discusses the implications for the reality of adiabatic geometric phases.

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.