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Journal of Physics Conference Series

Supercoherent states approach to the SUSY harmonic oscillator

Erik Díaz-Bautista, David J. Fernández C.

Published Jun 26, 2015• 0 citations
Publisher

Abstract

The nonlinear supercoherent states, associated with a nonlinear generalization of the Kornbluth-Zypman (KZ) supersymmetric annihilation operator (SAO) of the supersymmetric harmonic oscillator, will be studied. We discuss as well the Heisenberg uncertainty relation σ 2 x σ 2 p for a special case which will allow us to compare our results with those obtained for the KZ linear supercoherent states.

DOI
10.1088/1742-6596/624/1/012014

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The nonlinear supercoherent states, associated with a nonlinear generalization of the Kornbluth-Zypman (KZ) supersymmetric annihilation operator (SAO) of the supersymmetric harmonic oscillator, will be studied. We discuss as well the Heisenberg uncertainty relation σ 2 x σ 2 p for a special case which will allow us to compare our results with those obtained for the KZ linear supercoherent states.
This study generalizes the supersymmetric coherent states introduced by Aragone and Zypman in 1986. The Hamiltonian of the supersymmetric quantum harmonic oscillator leads to the definition of the generalized supersymmetric annihilation operators as a 3-parameter family. Their eigenstates are the generalized supercoherent states, which can be calculated explicitly for three relevant regions of parameter space. The uncertainty in position and momentum is discussed, with specific concentration on where the uncertainty is saturated, where it is bounded, and where it is unbounded.
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.
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.
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.