Path integral calculation of the Wigner function

Elementary Wigner function calculations of the infinite square well and Schroedinger cat states are presented as an introduction to the quasi-probability function. An entangled cat state is calculated and the Wigner function of the state is found. Properties of the entanglement of the state and the nature of its entanglement are found to be distinguishable by this distribution. This work is mostly concerned with obtaining the Wigner function via a path integral method, following a previously published technique. The method approximates the ground state Wigner function by finding the classical path associated with each point in phase space, assuming the P-function of the Hamiltonian of the system is able to be found. The imaginary part of action determines the phase of the path integral and depends on the geometry of the path; specifically the area which it encloses. An investigation into two systems, the Morse potential and the double well potential, was performed to try and find classical paths enclosing area and thus recreating the negative features of the exact Wigner function. The minimisation of the action found the classical path for each phase space point. This was performed numerically using tools created in Excel and Mathematica. In general, it was discovered that the classical paths did not enclose any area and therefore the Wigner function approximations were everywhere positive. The majority of those paths which were found to enclose some area produce a phase which is not large enough to change the sign of the path integral.