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arXiv:quant-ph/0612122 · Modern Physics Letters A

A q-OSCILLATOR WITH "ACCIDENTAL" DEGENERACY OF ENERGY LEVELS

Alexandre M. Gavrilik, A. P. Rebesh

Published Apr 19, 2007• 28 citations
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Abstract

We study main features of the exotic case of q-deformed oscillators (so-called Tamm–Dancoff cutoff oscillator) and find some special properties: (i) degeneracy of the energy levels En1 = En1 + 1, n1 ≥ 1, at the real value[Formula: see text] of deformation parameter, as well as the occurrence of other degeneracies En1 = En1 + k, for k ≥ 2, at the corresponding values of q which depend on both n1 and k; (ii) the position and momentum operators X and Pcommute on the state|n1> if q is fixed as [Formula: see text], that implies unusual uncertainty relation; (iii) two commuting copies of the creation, annihilation, and number operators of this q-oscillator generate the corresponding q-deformation of the non-simple Lie algebra su(2) ⊕ u(1)whose nontrivial q-deformed commutation relation is: [J+, J-] = 2J0q2J3-1 where [Formula: see text] and [Formula: see text].

DOI
10.1142/s0217732307022827

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We study main features of the exotic case of q-deformed oscillators (so-called Tamm–Dancoff cutoff oscillator) and find some special properties: (i) degeneracy of the energy levels En1 = En1 + 1, n1 ≥ 1, at the real value[Formula: see text] of deformation parameter, as well as the occurrence of other degeneracies En1 = En1 + k, for k ≥ 2, at the corresponding values of q which depend on both n1 and k; (ii) the position and momentum operators X and Pcommute on the state|n1> if q is fixed as [Formula: see text], that implies unusual uncertainty relation; (iii) two commuting copies of the creation, annihilation, and number operators of this q-oscillator generate the corresponding q-deformation of the non-simple Lie algebra su(2) ⊕ u(1)whose nontrivial q-deformed commutation relation is: [J+, J-] = 2J0q2J3-1 where [Formula: see text] and [Formula: see text].
Using the q, p-deformed oscillators as basic generating system, we obtain diverse classes (which form distinct sectors of functional continua) of novel versions of q-deformed oscillators, all of which share the property of "accidental" degeneracy within a fixed pair of energy levels Em = Em+1, m ≥ 1, occurring at the real deformation parameter fixed by an appropriate value q(m) that depends on m and on particular model. Likewise, the degeneracy E0 = Ek (where k ≥ 2) takes place, for properly fixed q = q(k), in most of those models. The formerly studied model of q-oscillator known as the Tamm–Dancoff cutoff deformed oscillator is contained in the continua as isolated special case.
In this paper, the [Formula: see text]-deformed bosonic Tamm–Dancoff oscillator algebra is considered. First, the quantum algebraic and representative properties of these deformed bosons are analyzed in detail. The representations of the [Formula: see text]-fermion algebra of Tamm–Dancoff type are also studied. Second, the high-temperature thermostatistical properties of a gas of Tamm–Dancoff type [Formula: see text]-fermions are investigated. The fermionic distribution function and the other important thermodynamic functions such as the entropy and the specific heat are derived in terms of the real deformation parameter [Formula: see text]. Finally, the time evaluation of a two-level atom in a Tamm–Dancoff oscillator trap interacting with a single-mode traveling light field is concisely discussed.
Abstract In this paper, we consider a system of the q-deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q-calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q-derivative, q-integral and q-exponential function are introduced. Using these properties, we construct a new formalism for the q-deformed quantum mechanics, which accordingly involve the q-adjoint operator and the q-Hermitian operator properties. We then derive the q-deformed Heisenberg relation, and develop the q-Hermitian momentum operator. The q-deformed Schrödinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q-deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.
The q-deformed bosonic Tamm–Dancoff oscillator algebra is considered. The coherent state of the q-deformed bosonic Tamm–Dancoff oscillator algebra is first constructed in detail. Second, the high-temperature thermostatistical properties of a gas of the Tamm–Dancoff oscillators are investigated. For high temperatures, the specific heat, the entropy and the equation of state for the system are derived in terms of the real deformation parameter q. The results obtained by the effects of Tamm–Dancoff type q-deformation show that they could be useful for further researches on understanding of mutual interactions between bosons and fermions in many-body quantum systems.
In this paper, we propose a full characterization of a generalized q-deformed Tamm-Dancoff oscillator algebra and investigate its main mathematical and physical properties. Specifically, we study its various representations and find the condition satisfied by the deformed q-number to define the algebra structure function. Particular Fock spaces involving finite and infinite dimensions are examined. A deformed calculus is performed as well as a coordinate realization for this algebra. A relevant example is exhibited. Associated coherent states are constructed. Finally, some thermodynamics aspects are computed and discussed.
Multidimensional two-parameter (q1, q2)-oscillators are of two kinds: one is invariant under the (ordinary) Lie group SU (d), whereas the other is invariant under the quantum group SU q(d) where q = q1/q2. It is shown that the q1 = q2 limit of both of these two-parameter oscillators coincide and give the q-deformed Newton oscillator which can be derived from the standard quantum harmonic oscillator Newton equation. The bosonic degeneracies of the excited levels of these oscillators are different for q1 ≠ q2, but coincide in the q1 = q2 limit.