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Shape-invariant potentials and an associated coherent state

Fukui, T, Aizawa, N

Published Jan 1, 1993• 69 citations
OpenAlex

Abstract

An algebraic treatment of shape-invariant potentials in supersymmetric quantum mechanics is discussed. By introducing an operator which reparametrizes wave functions, the shape-invariance condition can be related to a oscillator-like algebra. It makes possible to define a coherent state associated with the shape-invariant potentials. For a large class of such potentials, it is shown that the introduced coherent state has the property of resolution of unity.

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An algebraic treatment of shape-invariant potentials in supersymmetric quantum mechanics is discussed. By introducing an operator which reparametrizes wave functions, the shape-invariance condition can be related to a oscillator-like algebra. It makes possible to define a coherent state associated with the shape-invariant potentials. For a large class of such potentials, it is shown that the introduced coherent state has the property of resolution of unity.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.
Generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show that this generalized formalism is able (a) to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system, (b) to reproduce, as particular cases, results already known for shape-invariant systems (such as standard harmonic oscillator and Pöschl–Teller potentials as well as quantum deformed harmonic oscillator models) and (c) point to a formalism that provides a unified description of the different kind of coherent states for quantum systems, deformed or not deformed.
Algebraic approach to the integrability condition called shape invariance is briefly reviewed. Various applications of shape-invariance available in the literature are listed. A class of shape-invariant bound-state problems which represent two-level systems are examined. These generalize the Jaynes-Cummings Hamiltonian. Coherent states associated with shape-invariant systems are discussed. For the case of quantum harmonic oscillator the decomposition of identity for these coherent states is given. This decomposition of identity utilizes Ramanujan's integral extension of the beta function.
By introducing the shape-invariant Lie algebra spanned by the supersymmetric ladder operators plus the identity operator, we generate a discrete complete orthonormal basis for the quantum treatment of the one-dimensional Morse potential. In this basis, which we call the pseudo-number-states, the Morse Hamiltonian is tridiagonal. Then we construct algebraically the continuous overcomplete set of coherent states for the Morse potential in close analogy with the harmonic oscillator. These states coincide with a class of states constructed earlier by Nieto and Simmons [Phys. Rev. D 20, 1342 (1979)] by using the coordinate representation. We also give the unitary displacement operator creating these coherent states from the ground state.
We obtain the Robertson–Schrödinger uncertainty relation for shape-invariant systems and construct generalized Robertson intelligent states for these systems using an algebraic approach based on the supersymmetric quantum mechanics. Using the variances of generalized quadrature operators we study the coherency and squeezing properties, evaluating their dependence on different kinds of generalizations for shape-invariant systems with potential parameters related by a translation (Pöschl–Teller potential) and by a scaling (self-similar potential).