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Journal of Physics A Mathematical and General

The perturbed ladder operator method: closed form expressions of perturbed wavefunctions and matrix elements

N Bessis, Georges Bessis, Gisèle Hadinger

Published May 1, 1980• 10 citations
Publisher

Abstract

By mapping the perturbation scheme onto the ladder operator formalism, the field of application of the Schrodinger-Infeld-Hull factorisation method is enlarged. It is shown how, at each order of the perturbation, perturbed ladder operators can be constructed. Thus, without having to calculate explicitly either the excited unperturbed functions or any matrix element, one obtains analytical expressions of the perturbed eigenvalues in terms of the quantum numbers of the factorisable unperturbed problem. A three-terms recurrence relation, valid at any rank of the perturbation, is derived and leads to closed form expressions of the perturbed eigenfunctions. Consequently, a closed form expression of any matrix element on the basis of the perturbed eigenfunctions is easily obtained from the calculation of one unique particular integral.

DOI
10.1088/0305-4470/13/5/023

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By mapping the perturbation scheme onto the ladder operator formalism, the field of application of the Schrodinger-Infeld-Hull factorisation method is enlarged. It is shown how, at each order of the perturbation, perturbed ladder operators can be constructed. Thus, without having to calculate explicitly either the excited unperturbed functions or any matrix element, one obtains analytical expressions of the perturbed eigenvalues in terms of the quantum numbers of the factorisable unperturbed problem. A three-terms recurrence relation, valid at any rank of the perturbation, is derived and leads to closed form expressions of the perturbed eigenfunctions. Consequently, a closed form expression of any matrix element on the basis of the perturbed eigenfunctions is easily obtained from the calculation of one unique particular integral.
The perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of Hermite polynomials H k , when V(X)=b 2 X 2 + Sigma k C k H k (b 1 2/X). It is found that the use of a Hermite polynomials basis, together with the use of binomial coefficient functions in the quantum number, greatly simplifies the determination of the perturbed ladder and factorisation functions. Thus, one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation, without increasing intricacy. Thorough calculation has been given for a perturbing potential V(X) function even in X. As an illustrative application of the procedure, the resolution of the Schrodinger equation with a potential function V(X)=X 2 + lambda X 2 /(1+gX 2 ), g>0 is reinvestigated.
The theory of self-adjoint ladder operators is outlined, a comparison being made with the more usual type of ladder operator. It is shown how the method applies to the nnn-dimensional orbital angular momentum problem, for which it is found that the ladder operator may be expressed as a linear combination of the components of angular momentum, the expansion coefficients being the elements of a generalized spin algebra. Its relationship to the Infeld-Hull factorization method is discussed, the factorization appearing as part of a more general scheme. The physical significance of the ladder operators is emphasized through their application to the relativistic quantum theory of Dirac.
A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.
A nondegenerate perturbation theory is studied using the adiabatic-theorem formalism. A new factorization of the adiabatic time-evolution operator analogous to that obtained by Morita in his derivation of the linked-cluster theorem is given by carrying out a series of intermediate representations. This leads to a new expansion for the perturbed eigenvectors \ensuremath{\Vert}\ensuremath{\alpha}〉 and energies ${E}_{\ensuremath{\alpha}}$ which are analytically expressed with respect to two matrix series scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n). The nth term of each series (scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n)) includes perturbational contributions of power p, p\ensuremath{\ge}n. An explicit recurrence, scrV(n){\mathrm{scrV}}^{(n)}scrV(n)=V(${\mathrm{scrV}}^{n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$), E(n){E}^{(n)}E(n)=E(${\mathrm{scrV}}^{(n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.
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.