As a quasi-probability distribution function and a special representation of the density matrix,Wigner function is of great significance in physics.With the revision of nature and calculation method of Wigner function,the Wigner function of charged particles in the uniform magnetic field is obtained with the help of Star-eigenequation.
As Wigner quasi-probability distribution function,the Wigner function contains the entire information in the evolution of the quantum states in the phase space.The function also has great significance in physics.This paper introduces the definition and nature of the Wigner function,as well as the general expression.Then,it used the integral method given Wigner function of a one-dimensional infinitely deep square potential well.Finally,we discuss the characteristics of the distribution of the function and point out that it is a real function of the phase space.
As a quasi-probability distribution function in phase-space and a special representation of the density matrix, Wigner function has great significance in physics. In this paper, first, Wigner function’s characteristic and calculation approach are introduced. Then, with Star-eigen equation we obtain the Wigner function for three-dimensional harmonic oscillator. In the end, we discuss the method describing the interaction between phonons and electrons (or photons) and obtain the equation that transition probability satisfies in phase space.
Gauge-independent Husimi function (Q−function) of states of charged quantum particles in the electro-magnetic field is introduced using the gauge-independent Stratonovich-Wigner function, the corresponding dequantizer and quantizer operators transforming the density matrix of state to the Husimi function and vice versa are found explicitly, and the evolution equation for such function is derived. Also own gauge-independent non-Stratonovich Wigner function is suggested and its Husimi function is obtained. Dequantizers and quantizers for these Wigner and Husimi functions are given.
As is well known, there are several formulations of quantum mechanics. One of them, closely connected with the Weyl representation of operators [ 1 ] and Wigner functions [ 2 ], use functions on the phase space to represent both observables and quantum states. For working in this representation it is necessary to replace operators by their Weyl symbols and density matrices by Wigner functions [ 3 , 4 ]. After fundamental works of Weyl, Wigner and Kirkwood [ 1 , 2 , 5 ] properties of the representation were studied and used to calculate the properties of various physical systems (see [ 6–23 ]), including systems with magnetic field [ 15–23 ].