Principles of Quantum Mechanics · Problem 15.3.4

Addition of Angular Momenta — R. Shankar

Physics — Quantum Mechanics · magnetic-moment-two-angular-momenta

Problem

(1) Consider a system whose angular momentum consists of two parts

\mathbf{J}_1

and

\mathbf{J}_2

and whose magnetic moment is

\boldsymbol{\mu} = \gamma_1 \mathbf{J}_1 + \gamma_2 \mathbf{J}_2

In a state

|jm, j_1 j_2\rangle

show, using Eq. (15.3.19), that

\langle\mu_x\rangle = \langle\mu_y\rangle = 0

\langle\mu_z\rangle = m\hbar\left[\frac{\gamma_1 + \gamma_2}{2} + \frac{(\gamma_1 - \gamma_2)}{2}\frac{j_1(j_1+1) - j_2(j_2+1)}{j(j+1)}\right]

(2) Apply this to the problem of a proton (

g = 5.6

) in a

^2P_{1/2}

state and show that

\langle\mu_z\rangle = \pm 0.26

nuclear magnetons. (3) For an electron in a

^2P_{1/2}

state show that

\langle\mu_z\rangle = \pm\frac{1}{3}

Bohr magnetons.