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Communications in Theoretical Physics

q -Superoscillator Transformation Matrix and q -Supercoherent States

Hao San-Ru, LONG JUN-YAN, Guanghua Li

Published Jul 30, 1994• 0 citations
Publisher

Abstract

In this paper, we considered a q -superoscillator model and constructed a transformation matrix which transforms the usual algebra of q -boson and q -fermion operators of the q -superoscillator into a nonlinear shift automorphism algebra. The q -supercoherent states for the q -superoscillator model is introduced by utilizing the transformation matrix. They are compared with the coherent states defined as eigenstates of the annihilation operators. Matrix elements of the transformation matrix on the Fock space bases are evaluated as well.

DOI
10.1088/0253-6102/22/1/53

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In this paper, we considered a q -superoscillator model and constructed a transformation matrix which transforms the usual algebra of q -boson and q -fermion operators of the q -superoscillator into a nonlinear shift automorphism algebra. The q -supercoherent states for the q -superoscillator model is introduced by utilizing the transformation matrix. They are compared with the coherent states defined as eigenstates of the annihilation operators. Matrix elements of the transformation matrix on the Fock space bases are evaluated as well.
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.
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.
Starting from a faithful five-dimensional matrix representation of the group of two independent oscillators and applying the R-matrix method we generate some classes of deformed fermionic-bosonic quantum Hopf algebras. The corresponding Lie deformed superalgebras of type I--II, obtained by duality, are computed and a realization of generators of these deformed superalgebras are given in terms of the usual fermionic and bosonic creation and annihilation operators associated to the supersymmetric harmonic oscillator. Then, a generalized deformed annihilator is construted and their eigenstates are computed giving a new class of deformed coherent states.
We review some basic elements on k-fermions, which are objects interpolating between bosons and fermions. In particular, we define k-fermionic coherent states and study some of their properties. The decomposition of a Q-uon into a boson and a k-fermion leads to a definition of fractional supercoherent states. Such states involve bosonic coherent states and k-fermionic coherent states. We construct an Hamiltonian which generalizes the ordinary (or Z_2-graded) supersymmetric oscillator Hamiltonian. Our Hamiltonian describes a fractional (or Z_k-graded) supersymmetric oscillator for which the fractional supercoherent states are coherent states.