Wigner function for the orientation state

Abstract

We introduce a quantum phase space representation for the orientation state of extended quantum objects, using the Euler angles and their conjugate momenta as phase space coordinates. It exhibits the same properties as the standard Wigner function and thus provides an intuitive framework for discussing quantum effects and semiclassical approximations in the rotational motion. Examples illustrating the viability of this quasi-probability distribution include the phase space description of a molecular alignment effect.

Wigner representation of the rotational dynamics of rigid tops

“…Note that several known representations (e.g. [37,38]) allow a similar trick. However, their generalized parametric spaces are not singularity-free and suffer from the gimbal lock problem.…”

“…A variety of ways to extend the original Wigner quantization ansatz to the case of rotational dynamics were suggested and analyzed [27][28][29][30][31][32][33][34][35][36][37][38][39], but only a few of them are applicable to unrestricted rotations of 3-dimensional bodies. The early solutions of Refs.…”

“…[35,36], reduce the problem to the canonical case at the cost of extending the phase space by two artificial dimensions. A variant suggested in [37] allows to directly extract the most useful partial distributions but involves rather complicated quantum Liouville equations. Conversely, in the Nasyrov proposal [38], the equations for free symmetric and linear tops coincide with the classical ones at the expense of * dm.zhdanov@gmail.com † t-seideman@northwestern.edu complicated integro-differential form for dynamical equations and common observables in the general case.…”

Coherent distributions for the rigid rotator

“…In particular, for the angular momentum components ρ i and l i = k R ik ρ k , we get {ρ i , ρ j } = −ǫ ijk ρ k , respectively {l i , l j } = ǫ ijk l k . A local set of canonical coordinates on T * SO(3, R), related to Wigner distribution defined in [13], is presented in Appendix 1.…”

“…Proposed during the early days of the quantum theory [7] to explain the specific heats of diatomic gases, finite ground state-energy terms have been retrieved in the more recent years for the rigid rotator from geometrical considerations [8,9,10], and for the simple (linear) rotator by using reduced Wigner quasiprobability distributions [11]. Though, by adapting the known Wigner transform [12] from R 3 to SO(3, R), an additional "quantum potential" term was obtained [13].…”