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 Exercise [22.19] 
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Joined: 12 Jul 2010, 07:44
Posts: 154
Post Exercise [22.19]
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.


13 Oct 2012, 08:13
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