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The Journal of Chemical Physics

Perturbation Theory in Terms of Commutator Brackets

Gordon L. Goodman

Published Nov 15, 1965• 7 citations
Publisher

Abstract

Explicit formulas for changes produced by a perturbation of a quantum-mechanical system are given in terms of commutator brackets. Conditions that must be satisfied in order that these formulas be useful are discussed. Finally, it is shown how these new perturbation formulas are related to more usual ones in terms of energy denominators.
DOI
10.1063/1.1701507

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The quantum-mechanical state vector is not directly observable even though it is the fundamental variable that appears in Schr\"odinger'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.
The solution of the quantum-mechanical eigenvalue problem is discussed for cases when a series of approximate eigenfunctions is known. If these ``unperturbed'' states are divided into two classes, a perturbation formula is derived giving the influence of one class of states on the other in the final solution. The formula contains as special cases: (i) the Schrödinger-Brillouin formula for the eigenvalue of a nondegenerate state, (ii) a new simple formula for treating a class of degenerate states, and (iii) the splitting of the secular equation in cases where the system naturally consists of two independent parts in mutual interaction.