Introduction: Classical Mechanics Hiding in Quantum
Ehrenfest's theorem shows that quantum expectation values obey equations that look like classical mechanics. It's the bridge between the quantum and classical worlds.
The Time Evolution of Expectation Values
For any operator \(\hat{A}\):
\[\frac{d\langle\hat{A}\rangle}{dt} = \frac{1}{i\hbar}\langle[\hat{A}, \hat{H}]\rangle + \left\langle\frac{\partial\hat{A}}{\partial t}\right\rangle\]Ehrenfest's Results
Applying this to position and momentum:
\[\frac{d\langle x\rangle}{dt} = \frac{\langle p\rangle}{m}\] \[\frac{d\langle p\rangle}{dt} = -\left\langle\frac{\partial V}{\partial x}\right\rangle\]These look exactly like Newton's laws for the averages!
Worked Examples
Example 1: Free Particle
With \(V = 0\): \(\frac{d\langle p\rangle}{dt} = 0\), so \(\langle p\rangle\) is constant.
\(\langle x(t)\rangle = \langle x(0)\rangle + \frac{\langle p\rangle}{m}t\) — uniform motion!
Example 2: Harmonic Oscillator
With \(V = \frac{1}{2}m\omega^2 x^2\):
\[\frac{d\langle p\rangle}{dt} = -m\omega^2\langle x\rangle\]Combined with the first equation, \(\langle x\rangle\) oscillates sinusoidally.
The Quantum Connection
Ehrenfest's theorem explains why classical mechanics works for macroscopic objects. When wavepackets are narrow, \(\langle V'(x)\rangle \approx V'(\langle x\rangle)\), and the equations become exactly classical. Quantum mechanics contains classical mechanics as a limiting case.