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Exercise [22.26]
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Joined: 23 Aug 2010, 13:12
Posts: 33
Exercise [22.26]
Partial solution attached

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19 Sep 2010, 17:32

Joined: 12 Jul 2010, 07:44
Posts: 154
Re: Exercise [22.26]
Smith has shown above that any state where is an eigenstate of some operator where . But since this also holds true for all complex multiples of the operator or the eigenstate, we can conclude that every state is an eigenstate of the angular momentum operator along the (not necessarily unit) vector in real 3-dimensional space. (Recall , etc).

That completes part (i) of the problem.

For part (ii) we note that since is a representation space of SO(3), we can rotate the spin-up state to point in any direction (by applying the appropriate rotation from SO(3)). If represents the set of physically distinguishable eigenstates (i.e. we consider complex multiples of to represent the same thing), then there must be an element of corresponding to each possible rotated spin state . But is isomorphic to the Riemann sphere, i.e. to exactly the set of all possible directions in 3-space; so a topology preserving mapping , SO(3) must therefore be onto ; and hence every element of (and hence every spin state ) corresponds to some rotation of the spin-up state... i.e. to a spin eigenstate along some axis in 3-space.

02 Dec 2012, 09:23
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