Principles of Quantum Mechanics · Problem 20.12

The Dirac Equation — R. Shankar

Physics — Quantum Mechanics · dirac-equation-zitterbewegung

Problem

In the Heisenberg picture, the velocity operator for a Dirac particle is r˙=cα\dot{\mathbf{r}} = c\boldsymbol{\alpha}. Show that the eigenvalues of each component of α\boldsymbol{\alpha} are ±1\pm 1 (so the speed is always cc) and derive the equation of motion for α(t)\boldsymbol{\alpha}(t), showing that it contains an oscillatory term with frequency ω∼2mc2/ℏ\omega \sim 2mc^2/\hbar. This rapid oscillation is known as Zitterbewegung.

Browse the problem library