Perturbation theory for quantum-mechanical observables

The quantum-mechanical state vector is not directly observable even though it is the fundamental variable that appears in Schrödinger’s equation. In conventional time-dependent perturbation theory, the state vector must be calculated before the experimentally observable expectation values of relevant operators can be computed. We discuss an alternative form of time-dependent perturbation theory in which the observable expectation values are calculated directly and expressed in the form of nested commutators. This result is consistent with the fact that the commutation relations determine the properties of a quantum system, while the commutators often have a form that simplifies the calculation and avoids canceling terms. The usefulness of this method is illustrated using several problems of interest in quantum optics and quantum information processing.