Introduction to Quantum Mechanics · Problem 6.29

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · heisenberg_equations_of_motion

Problem

Differentiate Equation 6.72 to obtain the Heisenberg equations of motion

i\,\hbar\,\frac{d}{dt}\hat{Q}_H(t) = \left[\hat{Q}_H(t),\,\hat{H}\right] \tag{6.78}

(for

\hat{Q}

and

\hat{H}

independent of time). Plug in

\hat{Q} = \hat{x}

and

\hat{Q} = \hat{p}

to obtain the differential equations for

\hat{x}_H

and

\hat{p}_H

in the Heisenberg picture for a single particle of mass

m

moving in a potential

V(x)