Principles of Quantum Mechanics · Problem 14.4.5

Addition of Angular Momenta — R. Shankar

Physics — Quantum Mechanics · rotating-frame-effective-field

Problem

We would like to establish the validity of Eq. (14.4.26) when

\boldsymbol{\omega}

and

\mathbf{B}_0

are not parallel. (1) Consider a vector

\mathbf{V}

in the inertial (nonrotating) frame which changes by

\Delta\mathbf{V}

in a time

\Delta t

. Argue, using the results from Exercise 12.4.3, that the change as seen in a frame rotating at an angular velocity

\boldsymbol{\omega}

, is

\Delta\mathbf{V} - \boldsymbol{\omega}\times\mathbf{V}\Delta t

. Obtain a relation between the time derivatives of

\mathbf{V}

in the two frames. (2) Apply this result to the case of

\mathbf{l}

[Eq. (14.4.8)], and deduce the formula for the effective field in the rotating frame.