Introduction to Quantum Mechanics · Problem 4.42

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · lorentz_force_ehrenfest

Problem

(a) Using Equation 4.190 and the generalized Ehrenfest theorem (3.73), show that

\frac{d\langle \mathbf{r}\rangle}{dt} = \frac{1}{m}\langle(\mathbf{p} - q\mathbf{A})\rangle

. (4.192) Hint: This stands for three equations—one for each component. Work it out for, say, the

x

component, and then generalize your result. (b) As always (see Equation 1.32) we identify

d\langle\mathbf{r}\rangle/dt

with

\langle\mathbf{v}\rangle

. Show that

m\frac{d\langle\mathbf{v}\rangle}{dt} = q\langle\mathbf{E}\rangle + \frac{q}{2m}\langle(\mathbf{p}\times\mathbf{B} - \mathbf{B}\times\mathbf{p})\rangle - \frac{q^2}{m}\langle(\mathbf{A}\times\mathbf{B})\rangle

. (4.193) (c) In particular, if the fields

\mathbf{E}

and

\mathbf{B}

are uniform over the volume of the wave packet, show that

m\frac{d\langle\mathbf{v}\rangle}{dt} = q(\mathbf{E} + \langle\mathbf{v}\rangle \times \mathbf{B})

, (4.194) so the expectation value of

\mathbf{v}

moves according to the Lorentz force law, as we would expect from Ehrenfest's theorem.