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American Journal of Physics

Ladder operators in the Morse potential obtained from a related harmonic oscillator

R. Montemayor, Luis F. Urrutia

Published Jul 1, 1983• 16 citations
Publisher

Abstract

In this paper we explore a similarity transformation between the Morse oscillator and the two-dimensional harmonic oscillator. This transformation enables us to determine in a simple and straightforward way, all ladder operators for the Morse oscillator and their associated algebra, avoiding the use of the factorization method.
DOI
10.1119/1.13184

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The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.
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
Morse oscillator is one of the known solvable potentials which attracts many applications in quantum mechanics especially in quantum chemistry. One of the interesting results of this study is the generation of ladder operators for Morse potential. The operators are a representation of the shifting energy levels of the states exhibited by the wavefunction. From this result, we manipulate and deform the operators in such a way that it gives a noncommutative property to promote noncommutative quantum mechanics (NCQM). The resultant NC feature can be shown in the spatial coordinates and finally the Hamiltonian. In this study, we consider two-dimensional Morse potential where the ladder operators are in the form of the corresponding 2D Morse.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product