Franz Schwabl Quantum Mechanics 4th Edit · Problem 8.2

Spin

Physics — Quantum Mechanics · heisenberg-equations-harmonic-oscillator

Problem

Derive the Heisenberg equations of motion for the one-dimensional harmonic oscillator

H = \frac{1}{2m}p^2 + \frac{1}{2}m\omega^2 x^2 \;.

Compare with the classical equations of motion. Calculate the time dependence of the operators

a_{\mathrm{H}}

a^{\dagger}_{\mathrm{H}}

p_{\mathrm{H}}

, and

x_{\mathrm{H}}

. Determine

a_{\mathrm{H}}(t)

from the equation of motion and directly by use of the Baker--Hausdorff formula.