Relative-state formulation of quantum systems.

Abstract

A relative-state formulation of quantum systems is presented in terms of relative-coordinate states, relative-number states, and relative-energy states. The relative-coordinate states are used to describe quantum systems in position and momentum representations. The probability distribution is calculated in terms of the relative-coordinate states and is shown to be equivalent to the functional definition of the quantum probability in phase space. It is shown that a quantum-mechanical phase operator can be constructed in terms of the relative-number states without the well-known difficulties. The results are compared with those obtained by the Pegg-Barnett phase-operator formalism and the relations to various other phase-operator methods are also discussed. The energy-measurement and energy-probability distributions are discussed in terms of the relative-energy states. Furthermore, a relative-state formulation is developed in the Liouville space. A phase representation in the Liouville space is introduced to investigate the time evolution of quantum coherence. In the Liouville space a time operator is defined as a canonical conjugate of the time-evolution generator, but not the Hamiltonian energy operator. The relation to the internal time presented by Prigogine and Misra is discussed.