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Physical Review A

Perturbation theory for nonlinear time-independent Schrödinger equations

Péter R. Śurján, János G. Ángyán

Published Jul 1, 1983• 36 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.28.45

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American Journal of Physics•Sep 1, 1978•1 citation
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 \ensuremathΨ\ensuremath{\Psi}\ensuremathΨ. The formal results are applied to the case of the Onsager cavity model.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).
A modified version of Rayleigh-Schr\"odinger (RS) perturbation theory, which has been developed previously, can dispense with the usual RS restriction that the unperturbed Hamiltonian should be diagonal in the chosen basis. This modified RS (MRS) perturbation scheme is discussed in some detail and extended to the degenerate case. The new degenerate MRS perturbation theory remains based upon a rearrangement of the unperturbed and perturbed parts of the Hamiltonian. Basically, an unperturbed Schr\"odinger equation is thereby produced which may be satisfied trivially and identically for an arbitrary Hamiltonian. The zeroth-order approximation to the energy remains a free parameter of the degenerate version of the MRS formalism. The explicit MRS expressions for the higher-order correction terms also preserve a strict analogy with their counterparts in the nondegenerate formalism. What is new is an enormous increase in the applicability and universality of the whole approach. Thus, a priori, the unperturbed Hamiltonian may now be chosen so as to have all of its larger off-diagonal elements much closer to those of the full original Hamiltonian than is permitted in the standard RS approach. This is a particular advantage of the whole MRS approach. By testing the method on two nontrivial model Hamiltonians, the method is demonstrated to have considerably improved convergence properties over the standard RS scheme, and at relatively little extra cost.