John R Ramsden, as a comment on Motl’s blog, points us to an article by David Hestenes, Reading the Electron Clock, 0802.3227. There are two good reasons for looking at this paper, and the experiment that inspired it. First, we can use it as an excuse to discuss Hestenes’ electron theory, and what it looks like in the density operator language. Second, in a later post, we can discuss de Broglie’s matter waves.

**Zitterbewegung**

As the Wikipedia Zitterbewegung article states, Zitterbewegung [German for trembling motion] is an “interference between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median, with a circular frequency of , or approximately .” It was first noticed by Schoedinger, I think. It comes about when you compute the position operator, as a function of time, for an electron with momentum. The electron’s momentum p and mass m define a velocity v = p/m, and the position operator x ends up being given partly by vt = pt/m, but there is another term, an oscillatory term. [In quantum mechanics, the Hamiltonian H gives the energy, and setting H = mc^2, so the equation in the Wikipedia article looks like . Also, I'm mixing metaphors a bit between relativistic and non relativistic definitions of mass. Read the original articles for the correct derivation.] So long as you restrict to solutions of the Dirac equation which have only positive or negative energy, there is no Zitterbewegung, and this is how the frequency is usually ignored.

Hestenes is a long time advocate of The Zitterbewegung Interpretation of Quantum Mechanics. In his interpretation, Zitterbewegung (sometimes called zbw or zitter) is not due to interference between positive and negative frequencies but instead arises from the complex phase factor present in all quantum mechanics. Hestenes looks at the zbw as being associated with spin. The consequences are wide. He writes:

The essential feature of the zbw idea is the association of the spin with a local circulatory motion characterized by the phase factor. Since the complex phase factor is the main feature which the Dirac wave function shares with its nonrelativistic limit, it follows that the

. It follows that such familiar consequences of the Schroedinger theory as barrier penetration can be interpreted as manifestations of the zbw.Schroedinger equation for an electron inherits a zbw interpretation from the Dirac theory