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

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30 Sep 2010, 17:32

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

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

So it's not in fact the Pauli matrices themselves we have to exponentiate (as the text would seem to indicate), but the . (Or for rotation about an arbitrary axis, using your notation).

You've included (without really explaining why) the factor of , but forgotten the . Hence the bracketed terms in your expansions of and are not in fact equal to and 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.

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03 Nov 2012, 13:17
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