Quantum distribution functions for radial observables

The Wigner quasi-probability distribution function is a familiar tool to many working in quantum and atom optics [1]. It is primarily used in the classical-quantum correspondence where the appearance of positive and negative regions of the Wigner function gives easily understood information concerning the probability concentrations and quantum interferences present within the quantum state [2]. Typically one describes the Wigner function on a phase space which is labelled by cartesian coordinates of position and momentum. For physical systems which admit a two dimensional cylindrical symmetry, eg. trapped ultra-cold ions, bose condensates, etc., clearly a polar description of the Wigner function would be more natural. However, no such description has appeared in the literature. In this letter we show, in three stages, how a Wigner function for the radial observables can be constructed. This radial Wigner function could be reconstructed from experimental data much as recent experiments have reconstructed the cartesian Wigner function for the one dimensional motion of a trapped ion [3]. The angular parts of the complete four dimensional Wigner function are complicated by the imposition of singlevaluedness of the wavefunction under a rotation of 2π which cause the conjugate angular momentum to become discrete. We leave the angular part for a later work. The stages towards the construction of a radial-Wigner function proceed as follows: (1) a proper Wigner function possesses marginals which are true probability distributions for the observables whose eigenvalues label the Wigner function and thus the phase space axes. For a single degree of freedom a mere transformation of the cartesian position and momentum into polar form does not yield a proper Wigner function for the polar observables [4]. This is also true for higher dimensional phase space representations. Central to the problem is the correct specification of the radial momentum operator. By noting the symmetry action of the momentum on the half-infinite radius observable we construct a physical “conjugate” momentum P̂ ; (2) essential to the construction of the Wigner function is the existence of “point” operators Â(λ1, λ2), which obey, Tr[Â(λ1, λ2)Â (λ1, λ ′ 2)] = δ(λ1 − λ ′ 1)δ(λ2 − λ ′ 2) [5,11]. We find that the radial Â(r, P ), is not completely exponential in the radial position and momentum operators; (3) guided by the form of Â(r, P ) we make the coordinate transformation v̂ ≡ ln r̂. We find, [v̂, P̂ ] = ih̄, the Weyl algebra. After re-scaling the eigen-basis kets of v̂ to recover the standard resolution of unity we can easily construct the radial Wigner function, W (v, P ). This radial (or dilation) Wigner function gives the proper marginal probability distributions for v̂ and P̂ . We also note the existence of a dilaton ground state |0〉dilations, and calculate the wavefunction of this ground state in the r̂ basis. We finally calculate the radial Wigner function for the lowest Schwinger states |l, 0〉, (these are simultaneous eigenstates of energy and angular momentum), and briefly outline how, given a quantum state in a two dimensional harmonic Fock representation, one can construct the radial reduced density matrix, and from there the radial-Wigner function. Dirac in his textbook on quantum mechanics [6] introduced the following momentum, conjugate to the radial coordinate