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Physica Scripta

Algebraic approach to the Kratzer potential

Mohammad Reza Setare, Ebrahim Karimi

Published Dec 5, 2006• 101 citations
Publisher

Abstract

In this paper, the energy eigenvalues and the corresponding eigenfunctions are calculated for the Kratzer potential. Then we obtain the ladder operators for the one-dimensional (1D) and 3D Kratzer potential. Finally, we show that these operators satisfy the SU(2) commutation relation.
DOI
10.1088/0031-8949/75/1/015

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The ladder operators for the Kratzer–Fues oscillator have been derived within the algebraic approach. The method is extended to include the rotating Kratzer–Fues oscillator. For these operators, SU(2) Lie algebra has been constructed. The results obtained differ significantly from those recently derived by Setare and Karimi (2007 Phys. Scr. 75 90–3). We have shown that in their study the ladder operators and the solutions of the Schrödinger equations with the Kratzer potential have no physical meaning.
The eigenfunctions and eigenvalues of the three-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square interaction are obtained. A realization of the ladder operators for the wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρd/dρ with ρ = r 2 are evaluated. Another hidden symmetry explores the relations between the eigenvalues and eigenfunctions for substituting r→ ir. PACS number(s): 03. 65. Fd, 03. 65.Ge and 02. 20.Qs.
Abstract Taking a close look at the Infeld–Hull ladder operators for the Kratzer oscillator system, V(x) = [x2 + β(β − 1)x−2]/2, we deduce and explicitly construct energy‐raising and ‐lowering operators for the generalized Morse potential system V(z) = (Ae−4αz − Be−2αz)/2, through a canonical transformation that exists between the two systems. For the Morse potential system, we obtain a system of raising and lowering operators P±(n) (n = 0, 1, 2, 3, … , nmax) with the specific property that P±(n)Φn = c±(n)Φn±1, where Φn denotes the nth energy eigenfunction. While P−(0) annihilates the ground‐state Φ0, the operator P+(nmax), instead of annihilating the highest bound‐state Φ , actually knocks it out of the L2 space spanned by the discrete bound states and becomes inadmissible. Yet, raising and lowering operators P^\hat{P}P^ ± with proper end‐of‐spectrum behavior (i.e., P^\hat{P}P^ −|0〉 = 0 and P^\hat{P}P^ +|nmax〉 = 0) can be constructed in a straightforward way in the energy representation. We show that the operators P^\hat{P}P^ +, P^\hat{P}P^ −, and P^\hat{P}P^ 0 (where P^\hat{P}P^ 0 ≡ (1/2)[P^\hat{P}P^ +, P^\hat{P}P^ −]) form a su(2) algebra only if we restrict them to the (N − 1)‐dimensional subspace spanned by the lowest (N − 1) basis vectors, but not in the full (N + 1)‐dimensional space spanned by the discrete bound states [N ≡ nmax ≡ integral part of (1/2)(B/(2αA\sqrt{A}A​ ) − 1)]. Realization of this su(2) algebra in the position representation (when restricted to the (N − 1)‐dimensional subspace) is also given. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006