1.) Derive the expression of in -space. Prove is linear. Then, prove it is hermitian.
2.) Find the following commutators, showing two different ways for finding them:
(For first way use , and second way using properties of commutators).
3.) Show is an eigenfunction of . Determine the eigenvalue.
I can do this fairly easily in the "bra-ket" representation, but for some reason I'm getting a migraine from trying to rewrite it in terms of wavefunctions. I'll either get back to you on this, or I won't.
I'm going to leave the "hats" off. We know these are operators. (And besides, the x needs one, too!)
I'm sure one of the ways is
Now expand this out, do the expansion for , note that if then , etc.
I'm not sure if this is the other way you are thinking of. It may be shown that
and
If is an eigenvalue of the momentum operator p, then
where is some constant.
We know that
So solve
for
-Dan
I haven't been able to "derive" , but I proved is hermitian! Took a while:
We know it's hermitian if:
And my proof is:
We know that . So, we have:
Using integration by parts. Recall, it is . Let , and . The rest is trivial. Hence, we have:
Bring back into the integral and group terms (now it's negative, as we'll be taking the conjugate:
And thus we've shown that:
which is the definition of herminicity. Thus, the proof is complete.
But, my proof for showing it's linear needs some help:
Proof:
We can factor the constants out, since it'll just be another constant. But not sure how to show its linear from here.
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Struggling on 2 still, but I'm working on it.
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Found with your guidance. Lots of Quantum this week . Thanks topsquark.
Still need help with the linear part, but I worked on #2. It's quite messy, and took a long time, but I'm hoping you could help.
For the first method, we use the identity . Also, we use the fact that . Thus, we have:
Second method:
Obviously need help with it, but it looks somewhat right.
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Next one:
For the second one, the first method I use uses the multiplication property above, in addition to . Hence, we have:
Then of course we need to expand the other 2 brackets that I didn't do above at the end. But look how insanely long and complicated this is getting.
Second method:
Ahhhh!