Exercise [22.19]

deant · 13 Oct 2012, 08:13

Doing the matrix multiplication explicitly gives:

$L_1L_2 = - L_2L_1 = \frac{i\hbar}{2}L_3$

$L_2L_3 = - L_3L_2 = \frac{i\hbar}{2}L_1$

$L_3L_1 = - L_1L_3 = \frac{i\hbar}{2}L_2$

${L_1}^2 = {L_2}^2 = {L_3}^2 = \frac{\hbar^2}{4}\boldsymbol{I}$

From which the required commutation rules follow immediately.

We can relate the L's to quaternions by mapping:

$1 \leftrightarrow \boldsymbol{I} \; \; \; \; \; \; \boldsymbol{i} \leftrightarrow -\frac{2i}{\hbar}L_1 \; \; \; \; \; \; \boldsymbol{j} \leftrightarrow -\frac{2i}{\hbar}L_2 \; \; \; \; \; \; \boldsymbol{k} \leftrightarrow -\frac{2i}{\hbar}L_3$

Note that the bold i here is the quaternion i, whereas the non-bold i multiplying the L's is just the complex number i.

This is the mapping corresponding to $L_1=i\hbar l_1$ , etc. Another, equally good mapping can be obtained if we drop the minus signs, corresponding to quaternion rotations with the direction of rotation reversed.