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Physica Scripta

On an algebraic approach to the Kratzer oscillator

Damian Mikulski, Marcin Molski, Jerzy Konarski

Published Jul 27, 2009• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/80/02/025002

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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 coherent states for the Kratzer-Fues potential, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed using the algebraic approach. The method is extended to include the rotating Kratzer-Fues oscillator. Construction of such states is important in the theory of molecular interactions with coherent electromagnetic fields, for example, in calculation of the dynamic alternation of the refractive index due to the interactions of the laser pulse with the coherent vibration-rotational states of the diatomic molecule.
The ladder operators for the Goldman and Krivchenkov anharmonic potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The coherent states for the Goldman and Krivchenkov oscillator, which are the eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed within the framework of supersymmetric quantum mechanics. The constructed ladder operators can be a useful tool in quantum chemistry computations of non-trivial matrix elements. In particular, they can be employed in molecular vibrational–rotational spectroscopy of diatomic molecules to compute transition energies and dipole matrix elements.
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
By invoking a standard variable transformation x → y ≡ x 2 that connects the eigenvalue problems of the hydrogen atom system (Coulomb potential) and the Kratzer oscillator system , we obtain a system of ladder operators Ω nl ± ( l = 0,1,2, ... , n − 1) that connect the degenerate energy eigenstates (belonging to different l -values) of the hydrogen atom system. Using such quantum number specific operators as the precursors, we develop a general method of constructing quantum number independent ladder operators ; and then investigate the general conditions for these operators to form a realization of the su (2) algebra. In the process, we demonstrate that, in general, a finite-dimensional linear vector space carries a realization of the su (2) algebra. Applying the method to the case at hand, we show that for the hydrogen atom system the operators and form an su (2) algebra. The Hamiltonian is identified with the Casimir operator of the algebra, thereby revealing a dynamical SU (2) group symmetry of the hydrogen atom system.
We present exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse squared potential in arbitrary dimensions. The ladder operators for the radial wavefunctions are established by the factorization method. We find that those operators together with satisfy the commutation relation of the generators of an su(1, 1) algebra. Some comments are made on the general calculation formula for off-diagonal matrix elements and the recurrence relation among them. Also, we show that this anharmonic oscillator possesses a hidden symmetry between E ( r ) and E (i r ) by substituting r → i r .