Transition probabilities and transition rates in discrete phase space

We consider in what sense quantum tunnelling is associated with non-classical probabilistic behaviour. We use the Wigner function quasi-probability description of quantum states. We give a definition of tunnelling that allows us to say whether in a given scenario there is tunnelling or not. We prove that this can only happen if either the Wigner function is negative and/or a certain measurement operator which we call the tunnelling rate operator has a negative Wigner function...

Drawing inspiration from Dirac's work on functions of non commuting observables, we develop a fresh approach to phase space descriptions of operators and the Wigner distribution in quantum mechanics. The construction presented here is marked by its economy, naturalness and more importantly, by its potential for extensions and generalisations to situations where the underlying configuration space is non Cartesian.

We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave functions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features.

In this paper we will turn our attention to the problem of obtaining phase-space probability density functions. We will show that it is possible to obtain functions which assume only positive values over all its domain of definition.

We formulate continuous time quantum walks (CTQW) in a discrete quantum mechanical phase space. We define and calculate the Wigner function (WF) and its marginal distributions for CTQWs on circles of arbitrary length $N$. The WF of the CTQW shows characteristic features in phase space. Revivals of the probability distributions found for continuous and for discrete quantum carpets do manifest themselves as characteristic patterns in phase space.

We generalize classical statistical mechanics to describe the kinematics and the dynamics of systems whose variables are constrained by a single quantum postulate (discreteness of the spectrum of values of at least one variable of the theory). This is possible provided we adopt Feynman's suggestion of dropping the assumption that the probability for an event must always be a positive number. This approach has the advantage of allowing a reformulation of quantum theory in phas...

We analyze a known version of the discrete Wigner function and some connections with Quantum Iterated Funcion Systems. This paper is a follow up of "A dynamical point of view of Quantum Information: entropy and pressure" by the same authors.

Previously, an explicit solution for the time evolution of the Wigner function was presented in terms of auxiliary phase space coordinates which obey simple equations that are analogous with, but not identical to, the classical equations of motion. They can be solved easily and their solutions can be utilized to construct the time evolution of the Wigner function. In this paper, the usefulness of this explicit solution is demonstrated by solving a numerical example in which t...

This work is an extended version of the paper arXiv:0803.2669v1[math-ph], in which the main results were announced. We consider certain classical diffusion process for a wave function on the phase space. It is shown that at the time of order $10^{-11}$ {\it sec} this process converges to a process considered by quantum mechanics and described by the Schrodinger equation. This model studies the probability distributions in the phase space corresponding to the wave functions of...

We study the Wigner function for a quantum system with a discrete, infinite dimensional Hilbert space, such as a spinless particle moving on a one dimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase space construction, propose a meaningful definition of the Wigner function in this case, and characterize the set of pure states for which it is non-negative. We propose a measure of non-classicality for states in this system whi...