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ANZIAM Journal

Coherent states of para-Bose oscillators

S. N. Biswas, Thalanayar S. Santhanam

Published Oct 1, 1980• 143 citations
Publisher

Abstract

Abstract Para-Bose coherent states, defined as “displaced” ground states, are obtained using the differential operator representation for the annihilation operator.
DOI
10.1017/s0334270000002289

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In place of the usual commutation relation [â,â†]=1 we consider the generalized commutation relation characteristic of the para-Bose oscillators, viz, [â, Ĥ]=Ĥ where Ĥ is the Hamiltonian (1/2)(ââ†+â†â). The number states and the representation of various operators in the basis formed by these states are obtained. We then introduce the para-Bose coherent states defined as the eigenstates of â for this generalized case. We consider some of the properties of these coherent states and also show that the uncertainty product 〈 (Δq̂)2〉 〈 (Δp̂)2〉 in this case could be made arbitrarily small.
The construction of eigenstates of the square of annihilation operators for a two-mode para-Bose system is reported. Bose coherent states can be deduced from these eigenstates as a special case. These states are termed para-Bose coherent states. These states are degenerate. The expansion of the coherent states in terms of two-mode para-Bose energy eigenstates has been obtained and their salient properties are discussed. Also discussed is the uncertainty product of the square of position and momentum operators in the para-Bose coherent states for a two-mode system.
The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.