We compare the time evolution of the quantum-mechanical spatial probability density obtained by solving the time-dependent Dirac equation with its classical counterpart obtained from the relativistic Liouville equation for the phase-space density in a regime in which the dynamics is essentially relativistic. For a resonantly driven one-dimensional harmonic oscillator, the simplest nontrivial model system to perform this comparison, we find that, despite the nonlinearity induced by relativity, the classical ensemble description matches the quantum evolution remarkably well.
Grobe, Rainer; Su, Qichang; Peverly, P J.; and Wagner, R E., "Classical versus quantum dynamics for a driven relativistic oscillator" (2000). Faculty publications – Physics. 15.