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AIP conference proceedings

Trajectories of generalized quantum states for systems with finite discrete spectrum and classical analogs

Maia N Angelova, Anaelle Hertz, Véronique Hussin

Published Jan 1, 2012• 4 citations
Publisher

Abstract

In this paper, we construct generalized quantum states for systems with finite discrete spectrum and compare the behavior of those states with the classical analogs. Phase-space trajectories are analyzed in the classical and quantum cases. We focus on the following bounded anharmonic exactly solvable one-dimensional potentials: Morse, hyperbolic Pöschl-Teller and Rosen-Morse.

DOI
10.1063/1.4759390

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In this paper, we construct generalized quantum states for systems with finite discrete spectrum and compare the behavior of those states with the classical analogs. Phase-space trajectories are analyzed in the classical and quantum cases. We focus on the following bounded anharmonic exactly solvable one-dimensional potentials: Morse, hyperbolic Pöschl-Teller and Rosen-Morse.
We introduce a one-parameter-generalized oscillator algebra (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter κ. We define an (Hamiltonian) operator associated with and examine the degeneracies of its spectrum. For the finite (when κ < 0) and the infinite (when κ ⩾ 0) representations of , we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated-generalized oscillator algebra , where s denotes the truncation order. We construct two types of temporally stable states for (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of ). Two applications are considered in this paper. The first concerns physical realizations of and in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Pöschl–Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.
A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and P\"oschl-Teller, is given. The method, a priori, is potential independent and connects with earlier developed ones, including the oscillator-based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum-mechanical potential problems, such as the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.
Wave functions for the Coulomb, Morse, Rosen-Morse, and Eckart systems are expressed exactly and explicitly in terms of the corresponding classical trajectories. This is accomplished by a technique that maps previously obtained classical representations for wave functions of the three-dimensional isotropic harmonic oscillator to the wave functions for the present systems. By design, the classical behavior enters these expressions in a manner that allows them to be used as uniform semiclassical approximations for other systems. The construction of a hydrogen orbital from the corresponding classical orbits is illustrated for a particular case. The classical representation of the Eckart wave function is analyzed to understand how ordinary, real, classical trajectories that are confined to one side of a barrier are able to describe tunneling through the barrier.
I discuss in this paper the behaviour of the solutions of the so-called q-hyperbolic potentials, i.e. Pöschl–Teller-like and conditionally solvable potentials, in terms of the path integral formalism. The differences in comparison to the usual Pöschl–Teller-like potentials are investigated, including the discrete energy spectra and the bound state wave-functions. 1 Introduction. In this paper I want to discuss some specific generalisations of Pöschl–Teller related potentials. They are based on a q-deformation of the usual hyperbolic potentials, and are denoted by (we assume without loss of generality q> 0) Consequently we define
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.