Wigner Flow: The Quantum Analogue Of Phase Space Flow
    By News Staff | December 26th 2012 08:44 PM | 1 comment | Print | E-mail | Track Comments
    In classical physics, phase space trajectories give rise to flow-fields representing the dynamics of the system along its trajectories; they yield additional insight into a system's behavior.

    Quantum theory phase space trajectories do not exist because Heisenberg's uncertainty principle does not allow for the formation of sharply defined trajectories. But quantum physicists have not given up entirely on phase space, and so the study of the next best thing, the movement of quantum physics' phase space-based probability distributions, has actually boomed in recent years.

     UK-based physicist Ole Steuernagel from the University of Hertfordshire, alongside Dimitris Kakofengitis and Georg Ritter, have found that a new powerful tool they call ‘Wigner flow’ is the quantum analogue of phase space flow. Wigner flow provides information for quantum dynamics similar to that gleaned from phase space trajectories in classical physics. Wigner flow can be used for the visualisation of quantum dynamics. Additionally, and perhaps even more importantly, Wigner flow helps with the abstract analysis of quantum dynamics using topological methods.

    Steuernagel said, “Because trajectories are missing in quantum phase space, physicists did not pay much attention to the associated flow-fields, although these do exist. Now, our research shows that quantum phase space flow is well worth studying.” 

    Sophisticated schemes for the reconstruction of the most prominent of these distributions, ‘Wigner's function’, from experimental data, have set quantum phase space analysis on a firm footing. Yet, since quantum trajectory studies cannot be carried out, some of the power of established classical methods is missing.

    Researchers have been studying Wigner flow, which is based on the dynamics of Wigner's function, and have shown that it reveals new and surprising features of quantum phase space dynamics. It forms, for example, vortices that spin the ‘wrong’ way round and which appear in the ‘wrong’ part of phase space, when viewed from a classical physics standpoint. So, such dynamical patterns are manifestations of the quantum nature of the system.

    On top of such new riches the team has established the existence of a conservation law that reveals a new type of topological order for quantum dynamics. As an application they have shown that Wigner flow sheds new light on quantum tunnelling, the fundamental process that governs the workings of electronic computer circuits, and also the decay of radio-nuclides.

    Published in Physical Review Letters.


    It is amazing what a mathematical wonderland the 19th century mathematician William Rowan Hamilton’s
    p’s and q’s have led to.  It is little appreciated how much the work of the 19th century giants ploughed the field for quantum mechanics to develop.

    In regard to Phase Space — a direct development of Hamilton’s work — here is a little poem to enjoy:

    The Arnold Cat Map

    Eeuw-pop-eeeeuw! What’s that in the hall?
    Six little mices are having a ball,
    ’Coz the cat has been 
                                          which is quite undeserved.
    But wait!  Arnold’s Cat Map says volume’s conserved,
    In phase space at least – thank you, Arnold, for that,
    For we may now get back our poor pussy cat.
    Sorry, wee mices, you’ve just gotta go.
    Eeuw-pop-eeuw, Eeuw-poppô.

    (note: this little rhyme derives from the German children’s rhyme Eia, popeia, was raschelt im Stroh?, through a modification by Heinz Erhardt, who brought in the mice.)
    Robert H. Olley / Quondam Physics Department / University of Reading / England