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.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.