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Physical Review A

Group-theoretic intrepetation of the bound states of Morse oscillators

Carl E. Wulfman

Published Aug 1, 1996• 3 citations
Publisher

Abstract

Using operators of the invariance group of their time-dependent Schr\"odinger equations it is shown that the bound states of Morse oscillators are bases for discrete nonunitary irreducible representations (non-UIRs) of the groups $\mathrm{Sp}(2, R)$, SU(1,1), and SO(2,1). Hermiticity requirements and the dependence of the group generators on the dynamical variables determine the operator of an isometry which connects these non-UIRs of the noncompact invariance groups to UIRs of the compact groups SU(2) and SO(3) that have been used as spectrum-generating groups for Morse systems.

DOI
10.1103/physreva.54.r987

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Using operators of the invariance group of their time-dependent Schr"odinger equations it is shown that the bound states of Morse oscillators are bases for discrete nonunitary irreducible representations (non-UIRs) of the groups Sp(2,R)\mathrm{Sp}(2, R)Sp(2,R), SU(1,1), and SO(2,1). Hermiticity requirements and the dependence of the group generators on the dynamical variables determine the operator of an isometry which connects these non-UIRs of the noncompact invariance groups to UIRs of the compact groups SU(2) and SO(3) that have been used as spectrum-generating groups for Morse systems.
We propose a group-theoretical approach to the generalized oscillator algebra Aκ recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ ≥ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case κ < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Aκ in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices.
We propose a group-theoretical approach to the generalized oscillator algebra Aκ recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ ≥ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case κ < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Aκ in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices.
The bound-state solutions and the su(1,1) description of the d-dimensional radial harmonic oscillator, the Morse, and the D-dimensional radial Coulomb Schrödinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrödinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a d-dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a position-dependent mass background a well-known relationship in the context of constant mass.
In a recent paper (Aldaya V and Guerrero J 2005 J. Phys. A: Math. Gen. 38 6939 ) it is shown that the bound states of the Modified Pöschl–Teller potential should be described by the non-compact dynamical group SU (1, 1) instead of the usual compact group SU (2). Here we prove that SU (2) cannot be the dynamical group for a potential with a finite number of bound states on the basis of the Modified Pöschl–Teller potential and the Morse potential. This suggests that a quantum system with a continuum spectrum and a finite number of bound states should be described, both in the continuum and the discrete spectrum, by a non-compact dynamical group instead of a compact one.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.