Weyl–Wigner correspondence in two space dimensions

Abstract

We consider Wigner functions in two space dimensions. In particular, we focus on Wigner functions corresponding to energy eigenstates of a non-relativistic particle moving in two dimensions in the absence of a potential. With the help of the Weyl–Wigner correspondence we first transform the eigenvalue equations for energy and angular momentum into phase space. As a result we arrive at partial differential equations in phase space which determine the corresponding Wigner function. We then solve the resulting equations using appropriate coordinates.

Acknowledgements

It is a great honour and pleasure for us to dedicate this paper to Sir Peter Knight on the occasion of his 60th birthday. We look forward to many more years of wonderful scientific discussions and are grateful for his friendship.

This work has emerged from many fruitful discussions with I. Białynicki-Birula and D. Kobe. One of us (SV) is grateful to the Hungarian National Science Research Foundation (OTKA) project number T048324, and he thanks partial support from the COST P14 Action, reference code COST-STSM-P14-02507. Moreover, we appreciate the support of the Ministry of Science, Research and the Arts of Baden-Württemberg and the Landesstiftung Baden-Württemberg in the framework of the Quantum Information Highway A8 and the Center of Quantum Engineering.