M.Sc Thesis
M.Sc Student Ginzburg Dimitry Time Evolution of a Class of Quasi-Distribution Functions Department of Physics Professor Ady Mann

Abstract

One of the alternative formulations of Quantum Mechanics is the formulation by means of quasi distribution functions. This formulation allows calculating averages of observables in a way similar to classical mechanics - by integrating over phase space a function representing the observable multiplied by a quasi distribution function representing the state.

A quasi distribution function itself does not have the meaning of a distribution function. There exist an infinite number of quasi distribution functions - depending on the operator ordering we choose. For every quasi distribution function there is another way to represent the observables.

One of the important things is to calculate the evolution of the quasi distribution functions in time. In this thesis we are interested in evolution under a general quadratic Hamiltonian when additional damping is added. The damping is simulated by weak coupling to a heat bath. Ben Aryeh and Zoubi noticed that for this case, for the best known quasi-distribution function - the Wigner function, a technique that uses Lie algebras can be applied to find a convenient representation for the propagator of the function.

In this work we extend the technique presented by Zoubi and Ben Aryeh in order to propagate a more general class of quasi distribution functions under the same conditions. We show that the same technique can be used in order to propagate in time a class of quasi distributions that we call “general Gaussian quasi distribution functions”. The known Wigner, standard, anti-standard (Kirkwood-Rihaczek), normal and anti - normal quasi distribution functions are special cases of that “general Gaussian” quasi distribution function. We present two calculations. One for a case of quadratic Hamiltonian with damping; this calculation is derived from the equation of motion for the Wigner function. The second is a calculation for a quadratic Hamiltonian without damping. The second calculation uses the equation of motion for a general quasi distribution function which was presented by Leon Cohen in 1966. Of course the results of the second calculation are a special case of the first one. Finally, we show some examples of propagating different quasi distribution functions in time.