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arXiv:quant-ph/0108018 · Physical Review A

Perturbation theory for quantum-mechanical observables

J. D. Franson, Michelle M. Donegan

Published Apr 17, 2002• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.65.052107

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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.
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Modern computers have made possible the evaluation of higher order terms of perturbation series. Apart from the empirical work involving numerical calculations, much formal mathematical work has been done on the convergence properties of perturbation series and of Pade approximants which represent them. Some of the modern work in perturbation theory is reviewed in the context of traditional results from the theory of real and complex variables. The two major versions of time-independent perturbation theory, the Rayleigh-Schrodinger (RS) and Brillouin-Wigner (BW) theories, are compared and contrasted, and the alternative techniques for evaluating the terms in the energy series are discussed. The sum-over-states method, the differential equation method and the variational principle method are treated, with emphasis on their inter-relations.
The solutions of the time-independent Schr\"odinger equation for non-Hermitian (NH) Hamiltonians have been extensively studied and calculated in many different fields of physics by using L2{L}^{2}L2 methods that originally have been developed for the calculations of bound states. The existing non-Hermitian formalism breaks down when dealing with wave packets (WPs). An open question is how time-dependent expectation values can be calculated when the Hamiltonian is NH? Using the FFF-product formalism that was recently proposed by Moiseyev and Lein [J. Phys. Chem. 107, 7181 (2003)] we calculate the time-dependent expectation values of different observable quantities for a simple well-known study test case model Hamiltonian. We carry out a comparison between these results and those obtained from conventional (i.e., Hermitian) quantum mechanics (QM) calculations. The remarkable agreement between these results emphasizes the fact that in NH QM, unlike standard QM, there is no need to split the entire space into two regions, i.e., the interaction region and its surrounding. Our results open a door for a type of WP propagation calculations within the NH QM formalism that until now were impossible. In particular our work is relevant to the many different fields in physics and chemistry where complex absorbing potentials are introduced in order to reduce the propagation calculations to a restricted region in space where the artificial reflections from the edge of the numerical grid or box are avoided.
A number of quantum states form an orthonormal and complete basis set if they are normalized and mutually orthogonal and any quantum state can be expressed as a linear combination. In classical mechanics, any two physical variables are commutable. However, this property is no longer valid for physical operators. Commutation of operators is an important property of quantum mechanics. Although there are many representations of wave functions and observables connected by unitary transformations, it is useful to distinguish certain classes of representations, called pictures, which differ in the way the time evolution of the system is treated. The Schrodinger picture is one in which the operators are time-independent but the wave function is time dependent. The virial theorem relates the kinetic energy to its potential in an energy eigenstate.
This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.