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 Exercise [22.20] 
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Joined: 03 Jun 2010, 15:18
Posts: 136
Post Exercise [22.20]
Attached my solution

Exercise_22_20.pdf [61.42 KiB]
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30 Sep 2010, 17:32

Joined: 12 Jul 2010, 07:44
Posts: 154
Post Re: Exercise [22.20]
Sorry Roberto, but you've made a mistake here.

The generators of rotation around the 3 axes are the matrices l_1,l_2,l_3, and therefore a finite rotation through \theta is given by the exponentiation:

e^{\theta l_n}

So it's not in fact the Pauli matrices \boldsymbol{\sigma}_n themselves we have to exponentiate (as the text would seem to indicate), but the l_n = -\frac{i}{2}\boldsymbol{\sigma}_n. (Or -\frac{i}{2}\boldsymbol{\sigma} for rotation about an arbitrary axis, using your notation).

You've included (without really explaining why) the factor of \frac{1}{2}, but forgotten the -i. Hence the bracketed terms in your expansions of \boldsymbol{S_e} and \boldsymbol{S_u} are not in fact equal to cos\left(\frac{\theta}{2}\right) and sin\left(\frac{\theta}{2}\right) as you have indicated. Recall that the power series for sin and cos both have alternating + and - terms; that isn't the case here!!!

My solution (which is really just the same as yours but with these problems addressed) is attached below.

File comment: MS Word 2010 (original)
Exercise [22.20].docx [45.4 KiB]
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File comment: PDF version
Exercise [22.20].pdf [79.3 KiB]
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03 Nov 2012, 13:17
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