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arXiv:2302.04245

Considerations of the duality in generalized coherent states formalism

Dušan Popov

Published Feb 8, 2023• 0 citations
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Abstract

In the paper, a pair of dual operators is introduced with which a dual pair of generalized displacement operators is constructed. With these entities it is shown that the Barut - Girardello and Klauder - Perelomov generalized coherent states are dual states. The characteristics of these coherent states are constructed, separately and comparatively.

DOI
10.48550/arxiv.2302.04245

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In the paper, a pair of dual operators is introduced with which a dual pair of generalized displacement operators is constructed. With these entities it is shown that the Barut - Girardello and Klauder - Perelomov generalized coherent states are dual states. The characteristics of these coherent states are constructed, separately and comparatively.
We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting three different methods which are due to (i)(i)(i) Barut-Girardello, (ii)(ii)(ii) Gazeau-Klauder and (iii)(iii)(iii) Perelomov associated to Lie algebra su(1,1)su(1,1)su(1,1). Resolution of unity and overlapping properties are examined in each method.
Using the analytic representation of the so-called Gazeau–Klauder coherent states (CSs), we shall demonstrate that how a new class of generalized CSs, namely the family of dual states associated with theses states, can be constructed through viewing these states as temporally stable nonlinear CSs. Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the supersymmetric quantum mechanics (SUSYQM) techniques. Then, we have applied this method to some physical systems with known spectrum, such as Pöschl–Teller, infinite well, Morse potential and hydrogenlike spectrum as some quantum mechanical systems. Finally, we propose the generalized form of the Gazeau–Klauder CS and the corresponding dual family.
The two types of SU (1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU (1, 1) coherent states which takes on an intermediate position between the two dual types of SU (1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU (1, 1) coherent states are investigated for the realization of SU (1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU (1, 1) phase states are discussed for this realization of SU (1, 1). It is shown that no wavepackets corresponding to any of the SU (1, 1) coherent states can exactly preserve their shape during time evolution.
We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting the Gazeau–Klauder approach, where resolution of unity and overlapping properties are examined. Coherent states are found to be similar to those found for a particle trapped in a Pöschl–Teller potential of the trigonometric type. Some comparisons with Barut–Girardello and Klauder–Perelomov methods are noted.