Introduction to Quantum Mechanics · Problem 4.62

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · spin_matrices

Problem

Work out the spin matrices for arbitrary spin ss, generalizing spin 1/2 (Equations 4.145 and 4.147), spin 1 (Problem 4.34), and spin 3/2 (Problem 4.61). Answer: Sz=ℏ(s00⋯00s−10⋯000s−2⋯0⋮⋮⋮⋯⋮000⋯−s);\mathbf{S}_z = \hbar \begin{pmatrix} s & 0 & 0 & \cdots & 0 \\ 0 & s-1 & 0 & \cdots & 0 \\ 0 & 0 & s-2 & \cdots & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & \cdots & -s \end{pmatrix}; Sx=ℏ2(0bs00⋯00bs0bs−10⋯000bs−10bs−2⋯0000bs−20⋯00⋮⋮⋮⋮⋯⋮⋮0000⋯0b−s+10000⋯b−s+10)\mathbf{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & b_s & 0 & 0 & \cdots & 0 & 0 \\ b_s & 0 & b_{s-1} & 0 & \cdots & 0 & 0 \\ 0 & b_{s-1} & 0 & b_{s-2} & \cdots & 0 & 0 \\ 0 & 0 & b_{s-2} & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & b_{-s+1} \\ 0 & 0 & 0 & 0 & \cdots & b_{-s+1} & 0 \end{pmatrix} Sy=ℏ2(0−ibs00⋯00ibs0−ibs−10⋯000ibs−10−ibs−2⋯0000ibs−20⋯00⋮⋮⋮⋮⋯⋮⋮0000⋯0−ib−s+10000⋯ib−s+10)\mathbf{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -ib_s & 0 & 0 & \cdots & 0 & 0 \\ ib_s & 0 & -ib_{s-1} & 0 & \cdots & 0 & 0 \\ 0 & ib_{s-1} & 0 & -ib_{s-2} & \cdots & 0 & 0 \\ 0 & 0 & ib_{s-2} & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & -ib_{-s+1} \\ 0 & 0 & 0 & 0 & \cdots & ib_{-s+1} & 0 \end{pmatrix} where bj≡(s+j)(s+1−j)b_j \equiv \sqrt{(s+j)(s+1-j)}.

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