Introduction to Quantum Mechanics · Problem 4.33

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · spin_arbitrary_direction

Problem

Construct the matrix

S_r

representing the component of spin angular momentum along an arbitrary direction

\hat{r}

. Use spherical coordinates, for which

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

. (4.154) Find the eigenvalues and (normalized) eigenspinors of

S_r

. Answer:

\chi^{(r)}_+ = \begin{pmatrix} \cos(\theta/2) \\ e^{i\phi}\sin(\theta/2) \end{pmatrix}; \quad \chi^{(r)}_- = \begin{pmatrix} e^{-i\phi}\sin(\theta/2) \\ -\cos(\theta/2) \end{pmatrix}

. (4.155) Note: You're always free to multiply by an arbitrary phase factor—say,

e^{i\phi}

—so your answer may not look exactly the same as mine.