The Wigner function for the Dirac oscillator in spinor space is studied in this paper. Firstly, since the Dirac equation is described as a matrix equation in phase space, it is necessary to define the Wigner function as a matrix function in spinor space. Secondly, the matrix form of the Wigner function is proven to support the Dirac equation. Thirdly, by solving the Dirac equation, energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.
A precision measurment of inclusive electron scattering cross sections is carried out at Jefferson Lab in the quasi-elastic region for 4 He, 12 C, 56 Fe and 208 Pb targets. The longitudinal （R L ） and transverse （R T ） response functions of the nucleon need to be extracted precisely in the momentum transfer range 0.55 GeV/c≤ ｜ q ｜ ≤1.0 GeV/c. To achieve the above goal, a NaI （Tl） calorimeter is used to distinguish good electrons from background, including pions and low energy electrons rescattered from the walls of the spectrometer magnets. Due to a large set of kinematics and changes in HV settings, a number of calibrations are performed for the NaI （Tl） detector. Corrections for a few blocks of NaI （Tl） with bad or no signal are applied. The resolution of the NaI （Tl） detector after calibration reached δE/E^（1/2） ≈ 3% at E=1 GeV. The performance of the NaI （Tl） detector is compared with a simulation. The good calibration and background analysis for the NaI（Tl） detector are very important for the reduction of the systematic error of cross sections and the separation of R L and R T .
