Wigner Function And Application Of The Reciprocal Space Noncommutative (phase) Space

Wigner function is one of the most commonly used quasi-probability distribution functions in phase space. Especially for chemical physics problems, it renders clarified physical meanings, and is rich in features.Ever since the correlation between String Theory and Non-commutative Theory was revealed, researches on various physical problems in the non-commutative space have attracted great attention in the theoretical field of physics. In the string scale, the emergence of the non-commutative effects of spatial coordinates makes the space no longer commutative. In this thesis, our discussion is divided into two situations. One situation is the non-commutative space (NC space), where spatial coordinates do not commutate with each other while momentums still do. The other one is called non-commutative phase space (NC phase space), in which neither spatial coordinates normomentums commutate with their counterparts. This thesis deals with the Wigner function in non-commutative space and non-commutative phase space, respectively, mainly focused on the following two specific models:the charged harmonic oscillator in two-dimensional and the electronic motion in a magnetic field.There are two approaches to solve Wigner function:direct integration and solution of Wigner functions of the energy eigenstate equations. When it comes to the non-commutative space and non-commutative phase space, Bopp shift is employed in both cases to solve the Wigner function. As in NC space and NC phase space, the Schrodinger equation and the Wigner function of the energy eigenstate equations both contain star multiplication, which is equivalent to Bopp shift. Taking advantage of Bopp shift makes the process of solving the original complex simple and straightforward. For the importance of the Wigner function in the modern quantum measurement, the non-commutative quantum effects may be observed through the Wigner functions in experiments.