Introduction to Quantum Mechanics · Problem 7.31

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · angular_integration_identity

Problem

Let

\mathbf{a}

and

\mathbf{b}

be two constant vectors. Show that

\int (\mathbf{a} \cdot \hat{r})(\mathbf{b} \cdot \hat{r}) \sin\theta\,d\theta\,d\phi = \frac{4\pi}{3}(\mathbf{a} \cdot \mathbf{b})

(the integration is over the usual range:

0 < \theta < \pi

0 < \phi < 2\pi

). Use this result to demonstrate that

\left\langle \frac{3(\mathbf{S}_p \cdot \hat{r})(\mathbf{S}_e \cdot \hat{r}) - \mathbf{S}_p \cdot \mathbf{S}_e}{r^3} \right\rangle = 0,

for states with

\ell = 0

. Hint:

\hat{r} = \sin\theta\cos\phi\,\hat{i} + \sin\theta\sin\phi\,\hat{j} + \cos\theta\,\hat{k}

. Do the angular integrals first.