The statistical distribution for a system of q -deformed harmonic oscillators is studied and an explicit form of q -dependence of the distribution function is given. The physical meaning of the deformation parameter q is discussed.
Just as for the ordinary quantum harmonic oscillators, we expect the zero-point energy to play a crucial role in the correct high temperature behavior. We accordingly reformulate the theory of the statistical distribution function for the q-deformed boson oscillators and develop an approximate theory incorporating the zero-point energy. We are then able to demonstrate that for small deformations, ln q≪1, the theory reproduces the correct limits both for very high temperatures and for very low temperatures. The deformed theory thus reduces to the undeformed theory in these extreme cases.
The boson expansion method is applied to find the spectrum of a q-deformed harmonic oscillator. We use two different boson expansions, each of them including a deformation parameter, defined in terms of exponential and logarithmic functionals. The resulting Hamiltonians are bilinear forms of the transformed operators. Physical effects resulting from the deformation of the generators of the algebra are studied by comparing known finite-range potentials and the effective potentials obtained for each of the considered Hamiltonians.
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.