- #1
Chelsea S
- 12
- 0
Alright... So I'm in an 'introductory' Q.M class in college right now, it's the only one that this two-year college has, so I don't have an upper division Q.M Profs to talk to about this, and since my prof is equally confused, I turn to the internet.
Okay, so everyone knows that <ψ|Aψ> = <a>, where A is my observable, and <a> is my eigenvalue for the equation. Yay, verily, yay.
Now, we were going through a proof, and to begin he said what he was going to do, but didn't tell us why he was doing it (this is how he teaches, mysterious ways to keep our attention) anyway...
He gets to the end and the statement looks like this:
<f|g> = <ψ|ABψ> - <a><ψ|Bψ>
<f|g> = AB <ψ|ψ> - <a><b><ψ|ψ>
And everyone in the class says "wait, how in the hell did you not get eigenvalues for the AB combination in the wave function in the first set? (this one: AB<ψ|ψ>)
... And there was silence...
"Well, two years ago when I wrote these notes it made sense. I have no idea" he says.
So we discussed it for a little while, came up with some ideas as to why it probably couldn't. Physcially, I think (and please correct me if my thoughts are wrong here):
You have two particles that you want to measure *whatever* on in the same wave function. Your probability of measuring both of them at the same time where you want them to be would be less than if you had just a single one. That's what I would like to think anyway. So if indeed <ψ|ABψ> = <a><b>, the probability would be greater to find them both in the same wave function.
That's why I think it doesn't work.
I can't find anything anywhere about the nitty gritty of a rule that says that this can't work, and I can't come up with any math-like reasons or proofs as to why it can't work.
So..
Why does <ψ|ABψ> = AB and not <ψ|ABψ> = <a><b>
Okay, so everyone knows that <ψ|Aψ> = <a>, where A is my observable, and <a> is my eigenvalue for the equation. Yay, verily, yay.
Now, we were going through a proof, and to begin he said what he was going to do, but didn't tell us why he was doing it (this is how he teaches, mysterious ways to keep our attention) anyway...
He gets to the end and the statement looks like this:
<f|g> = <ψ|ABψ> - <a><ψ|Bψ>
<f|g> = AB <ψ|ψ> - <a><b><ψ|ψ>
And everyone in the class says "wait, how in the hell did you not get eigenvalues for the AB combination in the wave function in the first set? (this one: AB<ψ|ψ>)
... And there was silence...
"Well, two years ago when I wrote these notes it made sense. I have no idea" he says.
So we discussed it for a little while, came up with some ideas as to why it probably couldn't. Physcially, I think (and please correct me if my thoughts are wrong here):
You have two particles that you want to measure *whatever* on in the same wave function. Your probability of measuring both of them at the same time where you want them to be would be less than if you had just a single one. That's what I would like to think anyway. So if indeed <ψ|ABψ> = <a><b>, the probability would be greater to find them both in the same wave function.
That's why I think it doesn't work.
I can't find anything anywhere about the nitty gritty of a rule that says that this can't work, and I can't come up with any math-like reasons or proofs as to why it can't work.
So..
Why does <ψ|ABψ> = AB and not <ψ|ABψ> = <a><b>