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American Journal of Physics

On Degenerate Time-Independent Perturbation Theory

Carlos E. Solivérez

Published Jul 1, 1967• 9 citations
Publisher

Abstract

A formalism is presented in which both the degenerate and nondegenerate cases can be treated in the same fashion. Nonlinear integral equations are written for the perturbed wavefunctions, which lead to the usual formulas when solved by iteration. The method provides a simple way to set—up to any desired order—secular equations for the energy shifts and zeroth-order perturbed wavefunctions.

DOI
10.1119/1.1974198

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American Journal of Physics•Jul 1, 1983•0 citations
A formalism is presented in which both the degenerate and nondegenerate cases can be treated in the same fashion. Nonlinear integral equations are written for the perturbed wavefunctions, which lead to the usual formulas when solved by iteration. The method provides a simple way to set—up to any desired order—secular equations for the energy shifts and zeroth-order perturbed wavefunctions.
A special nondegenerate perturbation theory is developed for solving the "nonlinear Schr\"odinger equation" of the type $[\stackrel{^}{H}+\stackrel{^}{V}(\ensuremath{\Psi})]\ensuremath{\Psi}=E\ensuremath{\Psi}$, which arises, e.g., from the study of the environmental effects on molecules. The interaction operator is taken as $\stackrel{^}{V}(\ensuremath{\Psi})=\stackrel{^}{A}〈\ensuremath{\Psi}|\stackrel{^}{B}|\ensuremath{\Psi}〉$. The in vacuo problem $\stackrel{^}{H}\ensuremath{\Phi}=\ensuremath{\epsilon}\ensuremath{\Phi}$ is considered the zeroth-order description. For the perturbed energies and wave functions recursive formulas are derived at any order of the perturbation. The formulas obtained are compared to the usual Rayleigh-Schr\"odinger perturbation theory where $\stackrel{^}{V}$ is supposed to be independent of $\ensuremath{\Psi}$. The formal results are applied to the case of the Onsager cavity model.
There are few potentials V (r) for which the Schrödinger equation is exactly solvable. We must therefore develop approximation techniques to obtain the eigenvalues and eigenfunctions for potentials that do not lead to exactly soluble equations. One of these methods is the perturbation theory, which was developed both for time independent and time dependent phenomena. We will discuss the time independent perturbation theory. The formulation of the time independent perturbation theory is different for nondegenerate and degenerate states.
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.
Abstract Within the framework of the method of variation of constants, time-dependent perturbation theory is presented in a form that naturally eliminates the undesirable ‘secular ’ terms for a time-independent perturbation, permits an order-by-order calculation of the exact wavefunction for a ‘resonant’ harmonic perturbation and offers a simple route to establish the adiabatic hypothesis in quantum mechanics. The formulation rests on the introduction of a flexibility in the choice of the phase factors associated with the varying amplitudes, together with development of a perturbation procedure that closely resembles the Brillouin-Wigner formalism in the static case, and which exploits the flexibility of the phase factors at each order of the theory.