Solution of the free Dirac equation (Zitterbewegung)

The Dirac equation is the fundamental equation of relativistic quantum mechanics. While it improves on certain aspects of the Schrödinger equation it has solutions that are rather difficult to interpret.

This is a solution of the free Dirac equation in one space dimension. This is an equation for two-component wave functions. The plot shows both components of this "spinor-valued" wave function. The gray curve in the background gives the absolute value of the spinor (its square is often tentatively associated with a position probability density).

The initial function is a Gaussian function in both spinor components. The time evolution exhibits the phenomenon called "Zitterbewegung": The expectation value of the position oscillates around a mean value.

Note that despite the symmetry of the initial condition, the mean position drifts slowly to the right.

The reason for the Zitterbewegung is that the wave packet shown here is a superposition of positive- and negative-energy states. We refer to "Advanced Visual Quantum Mechanics" for a more detailed explanation.