Griffiths Introduction to Elementary Particles · Problem 4.23

Bound States — David J. Griffiths

Physics — QFT · spin-1-matrices

Problem

The extension of everything in Section 4.4 to higher spin is relatively straightforward. For spin 1 we have three states (

m_s = +1, 0, -1

), which we may represent by column vectors:

\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}

respectively. The only problem is to construct the

3 \times 3

matrices

\hat{S}_x

\hat{S}_y

and

\hat{S}_z

. The latter is easy: (a) Construct

\hat{S}_z

for spin 1. To obtain

\hat{S}_x

and

\hat{S}_y

it is easiest to start with the ``raising'' and ``lowering'' operators,

S_{\pm} \equiv S_x \pm iS_y

, which have the property

S_{\pm}|sm\rangle = \hbar\sqrt{s(s+1) - m(m \pm 1)}|s(m \pm 1)\rangle

(b) Construct the matrices

\hat{S}_+

and

\hat{S}_-

, for spin 1. (c) Using (b), determine the spin-1 matrices

\hat{S}_x

and

\hat{S}_y

. (d) Carry out the same construction for spin

\frac{3}{2}