For a closed quantum system the state operator must be a function of the Hamiltonian. When the state is degenerate, additional constants of the motion enter the play. But although it is the Weyl transform of the state operator, the Wigner function is not necessarily a function of the Weyl transforms of the constants of the motion. We derive conditions for which this is actually the case. The Wigner functions of the energy eigenstates of a two-dimensional isotropic harmonic oscillator serve as an important illustration.
The Wigner function W ( z , k ) for a model of interacting one-dimensional fermions confined to a harmonic trap is evaluated at zero temperature. The model considers two hyperfine states of the same fermionic species and treats the dominant interactions between the two using the bosonization method. Interactions substantially modify the shape of the Wigner function. Irrespective of the sign of the coupling constant, the Friedel oscillations inherent in the Wigner function are enhanced in the k -direction and suppressed in the z -direction.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.
In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form
W=Sp[ρW], where
ρ is the density matrix, and
W is the universal density matrix. The density matrix
ρ for each quantum system is different, and the universal matrix
W is the same for any quantum system. Thus, the matrix
W has a fundamental character. In the work, the elements of the universal matrix
W were found explicitly and their properties were investigated. The diagonal elements of the matrix
W are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix
W contain frequency oscillations responsible for dissipations in the quantum systems.