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arXiv:quant-ph/9804067 · Talk given at

Supercoherent States of Calogero-Sutherland Oscillator

Vladislav G. Bagrov, B. F. Samsonov

Published Apr 29, 1998• 0 citations
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Abstract

Supersymmetric quantum mechanical model of Calogero-Sutherlend singular oscillator is constructed. Supercoherent states are defined with the help of supergroup displacement operator. They are proper states of a fermionic annihilation operator. Their coordinate and superholomorphic representations are considered. The supermeasure on superunit disc which realizes the resolution of the unity is calculated. The cases of exact and spontaneously broken supersymmetry are treated separately. As an example the supersymmetric partners of the input Hamiltonian expressed in terms of elementary functions are given.

DOI
10.48550/arxiv.quant-ph/9804067

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Supersymmetric quantum mechanical model of Calogero-Sutherlend singular oscillator is constructed. Supercoherent states are defined with the help of supergroup displacement operator. They are proper states of a fermionic annihilation operator. Their coordinate and superholomorphic representations are considered. The supermeasure on superunit disc which realizes the resolution of the unity is calculated. The cases of exact and spontaneously broken supersymmetry are treated separately. As an example the supersymmetric partners of the input Hamiltonian expressed in terms of elementary functions are given.
A general approach is presented for constructing coherent states for supersymmetric systems. It uses Rogers's supermanifold formulation of supergroups to extend the group-theoretic method. Supercoherent states are explicitly obtained for the supersymmetric harmonic oscillator. They are shown to be eigenstates of the supersymmetric annihilation operator and to be minimum-uncertainty states. Two more-complex situations with extended physical supersymmetries are also considered: an electron moving in a constant magnetic field, and the electron-monopole system. The supercoherent states for these systems are found using super Baker-Campbell-Hausdorff relations and their interpretation is elucidated.
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The authors present the supercoherent states. They are introduced as the eigenstates of the supersymmetric annihilation operator of the quantum mechanical supersymmetric harmonic oscillator. They have a compact expression in terms of the standard (bosonic) coherent states. For each value of the complex parameter z there are two linearly independent orthogonal supercoherent states. One of them has pure fermionic character and is the more classical, while the other is fully supersymmetric (i.e. the mean value of the Klein operator when the system is set in this state vanishes). There are no purely bosonic supercoherent states. The supercoherent fermionic states saturate both the Heisenberg uncertainty relation and the new entropic uncertainty recently introduced by Deutsch (1983), while the supersymmetric ones almost (but not exactly) make it. They give a classical (or almost classical) mean value for the energy of the system and do not spread along their time evolution.
Coordinate atypical representation of the orthosymplectic superalgebra osp(2/2) in a Hilbert superspace of square integrable functions constructed in a special way is given. The quantum nonrelativistic free particle Hamiltonian is an element of this superalgebra which turns out to be a dynamical superalgebra for this system. The supercoherent states, defined by means of a supergroup displacement operator, are explicitly constructed. These are the coordinate representation of the known atypical abstract super group OSp(2/2) coherent states. We interpret obtained results from the classical mechanics viewpoint as a model of classical particle which is immovable in the even sector of the phase superspace and is in rectilinear movement (in the appropriate
We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q -deformed operators, we also give the q -deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q -supercoherent states associated with the q -deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q -supercoherent states are discussed as well.