Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Exercise [22.23]
Author Message

Joined: 23 Aug 2010, 13:12
Posts: 33
Exercise [22.23]
Partial Solution attached

Attachments:
Exercise22.23.doc [25 KiB]
18 Sep 2010, 20:24

Joined: 12 Jul 2010, 07:44
Posts: 154
Re: Exercise [22.23]
Smith, what makes you think that and are adjoint operators?

This is only equal to if and are both self-adjoint (Hermitian). It isn't obvious to me that this is the case.

Also, your solution is indeed only (very) partial, because you haven't established that all your eigenstates are also eigenstates of , nor that they all share the same eigenvalue j(j+1), nor why values of m below -j-1 or above j+1 can't be counted, nor why *all* values from -j to j should be, nor that there is only one eigenstate (and hence one dimension) per eigenvalue...

Furthermore, your solution is misleading, because it suggests that the dimension is 2j, corresponding to m values from -j+1 to j. It is indeed true that these are the values for which is nonzero, but by comparison the values of m for which is nonzero are -j to j-1; as the main text says, m actually runs from -j to j. The 2j dimension in the exercise statement is actually a typo - it ought to read:

Quote:
"...then the dimension is an integer 2j+1, where..."

...as the representation space is in fact (2j+1)-dimensional.

11 Nov 2012, 21:30
 Page 1 of 1 [ 2 posts ]