GR9677 #33



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Comments 
wittensdog 20091030 16:24:53  I can't help but comment that I think some of the explanations here are a little bit too complicated. This is a problem that should be no more than 10 seconds, and we don't need to think about bras and kets or anything like that.
For eigenstates of some quantity, which the spherical harmonics are for angular momentum, the probability of a measurement turning up a value associated with an eigenstate is just the square of the coefficient on that state. There are two states with l = 5, so we immediately just take the sum of the squares to get the total probability; 9/38 + 4/38 = 13/38.
In my opinion, any analysis more complicated than that is too much.   rawr 20090924 20:14:16  An even simpler way is just to realize that since two of the three states have l=5, the probability that it is in either of those two states is the same as the probability that it is NOT in the other state. So if you find the probability of the other state (25/38) and just subtract that from 1, you get (C).   tau1777 20081106 17:03:38  i get the ketbra manipulation part, but i still don't see how we take care of the fact that you're supposed to square this result.   cyberdeathreaper 20070114 11:56:41  Could someone expand on this calculation? I don't follow how to get it.
grae313 20071031 12:37:49 
If your wavefunction is given by then the probability of finding the particle in the nth state is . In this case there are two quantum numbers, but you still just square the 's that have the correct value of and add them together.

evanb 20080630 16:38:53 
Call each spherical harmonic  l, m >.
We know, because spherical harmonics are orthonormal, that
< ,  l, m > =
The operator we want to apply is
{ 1 if l = 5, 0 if l not = 5 }
Applying that operator to any ket  l m > yields zero if l isn't 5, and yields  5 m > if l IS 5.
Taking <  operator  > gives us the following:
(5 <1 1 + 3 < 5 1  + 2 <5 1 ) / * ( 3  5 1> + 2 5 1>)/$\sqrt{38}
Contracting, that yields, ( 3^2 <5 1 5 1> + 2^2 <5 15 1> ) / 38
Simplifying, (9+4)/38 = 13/38, which is answer (C).

 

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