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Collection of Czechoslovak Chemical Communications

A Class of Exactly Solvable Schrödinger Equations

Jacek Karwowski, Lech Cyrnek

Published Jan 1, 2005• 13 citations
Publisher

Abstract

An algebraic approach to solving a class of one-particle Schrödinger equations is presented. As an example, quasi-exact solutions of the eigenvalue problem of a Hamiltonian describing two interacting particles confined in a parabolic well are obtained. This example constitutes a unification and a generalization of several models known in the literature, as the ones of Taut (Phys. Rev. A 1993, 48, 3561) and of Samanta and Ghosh (Phys. Rev. A 1990, 42, 1178). Two confined particles interacting by Coulomb forces and the nuclear motion of a diatomic molecule are discussed as practical implementations.

DOI
10.1135/cccc20050864

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An algebraic approach to solving a class of one-particle Schrödinger equations is presented. As an example, quasi-exact solutions of the eigenvalue problem of a Hamiltonian describing two interacting particles confined in a parabolic well are obtained. This example constitutes a unification and a generalization of several models known in the literature, as the ones of Taut (Phys. Rev. A 1993, 48, 3561) and of Samanta and Ghosh (Phys. Rev. A 1990, 42, 1178). Two confined particles interacting by Coulomb forces and the nuclear motion of a diatomic molecule are discussed as practical implementations.
The number of exactly solvable eigenvalue problems has been rapidly increasing since the advent of quantum mechanics and there are many efficient ways to treat them. As a general rule, it is possible to write exactly solvable abstract eigenvalue equations in terms of the generators of finite-dimensional Lie algebras and solve them by means of purely algebraic methods. Matrix elements are commonly necessary for the application of perturbation theory and variational principles, among other approximate methods. The chapter discusses examples of three-dimensional Lie algebras with well-known quantum-mechanical realizations. Typically, exactly solvable quantum-mechanical problems are either onedimensional models or lead to separable equations in many dimensions. In the coordinate representation both cases reduce to ordinary differential equations of second order that determine the eigenvalues and separation constants compatible with the physical boundary conditions. In its purest form, a Lie algebraic method is representation-independent. Extensive application of algebraic methods to selected physical problems is already available elsewhere.
Exactly solvable model of two Rydberg-dressed atoms moving in a quasi-two-dimensional harmonic trap is introduced and its properties are investigated. Depending on the strength of inter-particle interactions and the critical range of the potential, the two-particle eigenstates are classified with respect to the excitations of the center-of-mass motion, relative angular momentum, and relative distance variable. Having these solutions in hand, we discuss inter-particle correlations as functions of interaction parameters. We also present a straightforward prescription of how to generalize obtained solutions to higher dimensions.
We present numerically exact solutions of the time-independent Schrödinger equation for the following system: two particles of the same charge, repelled by the Coulomb force, confined to a one-dimensional infinite well. The eigenfunctions are expanded in a basis set of product delta functions; the expansion allows the removal of the Coulomb potential’s singularity. We report and discuss our findings regarding correlated behavior in the lowest energy states of a well of length 3 bohr (1 bohr=1 Bohr radius≈0.52 Å).
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.
The Pauli–Schrödinger equation for a spin 1/2, neutral particle with a nonvanishing magnetic moment μ0, interacting with an external scalar potential V and a static magnetic field B, both functions of only one of the coordinates, is solved exactly for four different choices of the potential and the field. By choosing in the examples the coordinate y, we present these solutions in the following cases: (i) V( y)=0, B( y)=(B0sin κy, 0, B0cos κy) where B0 and κ are two arbitrary constants. (ii) V( y)=λ′y2, B( y)=‖αy‖(cos 2κy, 0, sin 2κy) where 6* λ=ℏ2/2m and α,κ arbitrary constants and λ′=λ(μ0/2κλ)2. (iii) V( y)=λ′(c̄1tan αy+c̄2cot αy)2, B( y)=B( y) (sin (2κy+2δ(y)), 0, cos (2κy+2δ( y)) where B2( y)=(c̄1tan αy+c̄2cot αy)2 (+α2/4κ2)(c̄1sec2αy−c̄2csc2αy)2 and tan 2δ( y)=(2κ/λ)((c̄1tan αy+c̄2cot αy) /(c̄1sec2αy−c̄2csc2αy)); c̄1, c̄2, α, and κ are arbitrary constants. (iv) V( y)=λ′(ā tanh z+b̄)2 where z =[(y−cd)/d], B2( y)=(ā tanh z+b̄)2+(ā2/4κ2d2)sech2z, tan 2δ( y)=(2κd/ā)((ā tanh z+b̄)/sech2z); ā, b̄, c, d, and κ are arbitrary constants restricted only by (μ0/2κλ)ā2+ā/d>0. The functions B( y) and δ( y) define the vector B( y) as in (iii). Our method of solution is based on the familiar factorization technique for solving the Schrödinger eigenvalue equation. Several interesting physical results of these solutions are discussed.
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer, and modified Kratzer potentials. We found the exact bound state solutions of the two--dimensional Schrödinger equation with Kratzer--type potentials and present analytical expressions for the eigenvalues and eigenfunctions. The eigenfunctions are given in terms of the associated Laguerre polynomials.