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Journal of Physics A Mathematical and General

Arbitrary-order parasupersymmetric coherent states of quantum harmonic oscillator

Hossein Fakhri, Mirenayatollah Bahadori

Published Sep 29, 2000• 2 citations
Publisher

Abstract

The eigenstates of arbitrary-order parasupersymmetric Hamiltonian p corresponding to a particle with spin p /2 in the presence of a harmonic oscillator potential and constant magnetic field directed along the z -axis are constructed in terms of eigenstates of a one-dimensional harmonic oscillator. Also, parasupersymmetric coherent states with degenerate multiplicity of an ad hoc bosonic annihilation operator of parasupersymmetric eigenstates of the Hamiltonian mentioned above are calculated.

DOI
10.1088/0305-4470/33/40/311

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The eigenstates of arbitrary-order parasupersymmetric Hamiltonian p corresponding to a particle with spin p /2 in the presence of a harmonic oscillator potential and constant magnetic field directed along the z -axis are constructed in terms of eigenstates of a one-dimensional harmonic oscillator. Also, parasupersymmetric coherent states with degenerate multiplicity of an ad hoc bosonic annihilation operator of parasupersymmetric eigenstates of the Hamiltonian mentioned above are calculated.
Coherent states are constructed in p=2-parasupersymmetric quantum mechanics in connection with the two relative para-Bose and para-Fermi sets of trilinear structure relations. They are called parasupercoherent states. Parasupersymmetric operators [such as the Hamiltonian and (two) annihilation operators] are introduced and discussed. In particular, the parasuperspectrum is determined and compared with the recent results obtained by Rubakov and Spiridonov [Mod. Phys. Lett. A 3, 1337 (1988)]. The superalgebra contents subtended by osp(2/2) and osp(3/2) are analyzed and exploited in order to get constants of motion in both contexts through parallel properties on the supersymmetric harmonic oscillator.
\n We have formulated a model of a complex (two-dimensional) quantum harmonic oscillator. All dynamical physical variables are expressed in terms of the creation and annihilation operators, viz., $a_{z}, \\bar{a}_{z}\\ {\\rm and} \\ a_{\\bar{z}}, \\bar{a}_{\\bar{z}} $. The Hamiltonian of the system is $H_{z\\bar{z}} = (\\bar{a}_{z}a_{z} + 1) + \\omega L_{z} $, where ω is the oscillator frequency and $L_{z} = \\frac{\\hbar }{2}\\,(\\bar{a}_{\\bar{z}} a_{\\bar{z}}-\\bar{a}_{z}a_{z}) $ is the orbital angular momentum. The oscillator is found to be described by a conserved orbital angular momentum (Lz) besides energy. While the ground-state wave function is real, all excited states are complex and degenerate. The oscillator in these states carry a quantum of charge of $e^{{\\ast}} = \\frac{n}{2n\\pm1} e $. These degenerate wave functions are eigenstates of the orbital angular momentum with eigenvalues nℏ and −nℏ, where h=2πℏ is the Planck's constant and n=1, 2, … . The two wave functions are degenerate with energy En=(n+1)ℏω. The comparison with Landau level reveals that in the presence of the magnetic field, B, where ω is equal to the cyclotron frequency, the current moment is quantized and is proportional to the square root of the magnetic field, i.e., ${\\cal I}_{n}\\propto n e \\sqrt{B} $.\n
In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.