Schroedinger described the rapid oscillation of the electron electric dipole moment and velocity in the Dirac theory as "zitterwebegung," or jitter motion. In his 1930 paper he gave the zitterbewegung frequency as 2 m c^2 / h. Here, m is the electron mass, c is the speed of light and h is the Planck constant.
The zitterbewegung frequency f_z = 2 m c^2 / h can be related to the de Broglie wavelength as follows.
If the electron is moving with speed expressed as a fraction of the speed of light as b = v/c, then time dilation reduces the observed zitterbewegung frequency by the Lorentz factor of L=1/(1-b^2)^(1/2). Supposing b << 1, then L ~ 1 + b^2/2. The difference between the zitter frequency in the electron rest frame and as observed is f_d ~ (b^2/2)f_z = v^2 m / h = v p / h, where p = m v. The distance traveled by the electron in one period of an oscillation at the difference frequency is then d = v / f_d = h / p, that is, the de Broglie wavelength of the electron.
I argue that all that needs to be assumed to find a physical interpretation for this distance is that the electron is a classical point charge and that classical electrodynamics and the Dirac theory are both true. Then a point charge moving as required to account for the observed intrinsic angular momentum of the electron and such that the electric dipole moment oscillates at the zitterbewegung frequency must magnetically interact with another similarly-moving point charge, and the magnetic force between them is sinusoidally modulated with frequency f_d. Further, the magnitude of the oscillating magnetic force between the two charges is equal to that of the Coulomb force between two stationary or relatively slowly-moving charges.
The longer version of this is in my paper now published by Foundations of Physics here: http://link.springer.com/article/10.1007/s10701-016-9990-1 The arxiv version is here: http://arxiv.org/abs/1409.8271