The Wigner function of a q-deformed harmonic oscillator model

Abstract

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 $h\to 0$ ($q\to 1$), 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 $q\ll 1$, 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.

16 pages, 24 EPS figures, uses IOP style LaTeX, some misprints are correctd and journal-reference is added

References in corpus (6)

• Quantum state reconstruction of the single-photon Fock state
• Quantum homodyne tomography of a two-photon Fock state
• q-deformed structures and generalized thermodynamics
• X rays test the Pauli exclusion principle
• Q-boson interferometry and generalized Wigner function
• The Wigner function associated to the Rogers-Szego polynomials

Cited by in corpus (4)

• The Wigner distribution function for the one-dimensional parabose oscillator
• The Wigner function of a semiconfined harmonic oscillator model with a position-dependent effective mass
• Transition to sub-Planck structures through the superposition of q-oscillator stationary states
• The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass