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Journal of the Nigerian Association of Mathematical Physics

Logarithmic perturbation theory: Applications and limitations

GY Ndefru, Salihu S Duwa

Published May 22, 2008• 0 citations
Publisher

Abstract

The time independent, non-degenerate standard perturbation theory is compared the alternate treatment of perturbation theory called logarithmic perturbation theory (LPT). For determining the non-degenerate ground state the LPT is, in principle, easier to apply than standard perturbation theory. This is because, as opposed to the standard perturbation method which requires the knowledge of the complete set of eigenvalues and eigenfunctions of the unperturbed system, for the LPT one only needs to know the ground state wave function of the unperturbed system, the energy correction to the next lower order and some easily computable coupling constants. However, in reality, the LPT is a simpler method to apply when the ground state wavefunction is exponential in nature. But as shown here for trigonometric unperturbed wave function the LPT leads to integrals which have no analytical solutions thereby making LPT more difficult and less accurate method than the standard perturbation approach. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 595-600

DOI
10.4314/jonamp.v10i1.40183

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The time independent, non-degenerate standard perturbation theory is compared the alternate treatment of perturbation theory called logarithmic perturbation theory (LPT). For determining the non-degenerate ground state the LPT is, in principle, easier to apply than standard perturbation theory. This is because, as opposed to the standard perturbation method which requires the knowledge of the complete set of eigenvalues and eigenfunctions of the unperturbed system, for the LPT one only needs to know the ground state wave function of the unperturbed system, the energy correction to the next lower order and some easily computable coupling constants. However, in reality, the LPT is a simpler method to apply when the ground state wavefunction is exponential in nature. But as shown here for trigonometric unperturbed wave function the LPT leads to integrals which have no analytical solutions thereby making LPT more difficult and less accurate method than the standard perturbation approach. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 595-600
We discuss an alternative method of performing coupling constant perturbation expansions in nonrelativistic quantum mechanics. This method, called logarithmic perturbation theory, yields new expressions for any-order corrections En to an unperturbed bound-state energy which do not involve cumbersome sums over intermediate unperturbed states. In one dimension, these corrections En can be evaluated using a simple explicit form containing a small number of integrals. In more than one dimension the approach is systematic but computations require the solution of well-defined partial differential equations. For order n=2 this equation is identical to that appearing in the method of Dalgarno and Lewis. Numerous illustrative examples are presented.
We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schrödinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime.
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In many situations in physics, the Hamiltonian of a given system is so complicated that the solution to the stationary Schrodinger equation is practically impossible or very difficult. Therefore, some approximation methods are required. This chapter presents the time-independent perturbation theory, the procedure of finding corrections to non-degenerate eigenvalues and eigenvectors to a part of the Hamiltonian of a given system. The perturbation theory is appropriate when the Hamiltonian can be split into two parts. The perturbation theory can be applied to analyze the quantum properties of particles trapped in two closely coupled potential wells. By closely coupled wells, it is meant that these two wells are separated by a barrier. This is a typical situation in the studies of quantum dynamics of Bose–Einstein condensates. Using the first-order perturbation theory, one can find the eigenvalues and eigenvectors.
The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.
Exactly solvable models in quantum mechanics are very rare.However, perturbation theory is a powerful tool to research the complex quantum systems and is widely used in theoretical physics and chemistry.For the particle weakly perturbed by the surrounding environment, its energy eigenstate and eigenvalue are separately calculated up to the first-order and second-order perturbation approximations in the undergraduate textbook of quantum mechanics.Whereas in reality, we have to calculate the high-order perturbation correction terms occasionally.Along the same lines of the low-order perturbation correction terms of energy eigenvalues and eigenstate for stationary state perturbation theory in the non-degenerate case, the third-order and fourth-order perturbation correction terms are strictly attained in this paper.Moreover, it points out the essence of perturbation theory and provides its derivation ideas.Students are no longer afraid of tedious derivation and can derive any-order perturbation correction.These results can serve as an extension and supplement for the quantum mechanics textbooks.Furthermore, it is very helpful for students to deeply understand quantum perturbation theory.