How does the momentum operator prove its role as the generator of translation?

[csutton1]

Does anyone have a momentum operator proof? The book I'm using is skipping a lot of steps .

 

[siddharth]

Could you be more specific? Proof of what?

 

[csutton1]

sorry i re-read my post and realized it was really vague. The momentum operator used to determine the momentum of a particle described by schrodingers wave equation.

 

[siddharth]

sorry i re-read my post and realized it was really vague. The momentum operator used to determine the momentum of a particle described by schrodingers wave equation.

Yes, what about it? I don't understand what proof you have in mind. Maybe, if you post the exact problem statement, and where you are stuck, it'll be easier to help.

 

[malawi_glenn]

I am also stunned, are you saying that you want a proof that

$$-i\hbar \nabla$$

gives the momentum of a state?

Be specific, and write down for us where you get stucked.

 

[JK423]

There is no *proof*, but there is a "classical" way to calculate it. But still it`s not a *proof*.
Assume that this relation for the mean values is valid:
<P>=m*d<x>/dt.
Then substitute <x>=(Ψ,xΨ), do the math and you`ll get the well known result:
<P>=(Ψ,pΨ), where p is the momentum`s operator.

 

[malawi_glenn]

No JK423, that is not what you do. You want to show that the p operator is the generator of translations and that it fulfills certain commutation relations.

 

[JK423]

No JK423, that is not what you do. You want to show that the p operator is the generator of translations and that it fulfills certain commutation relations.

I mentioned a way to calculate momentum`s operator in {x} representation from a known classical equation. What`s wrong about that?
What do you mean by
"You want to show that the p operator is the generator of translations and that it fulfills certain commutation relations" ?
You can show these these things without knowing what operator p equals to?

 

[malawi_glenn]

Well what you wrote is ok :-)
but is not a proof, as you said.



but I know what p equals, so I can proove that it is the generator of translation and fulfills certain commutator relations. I did not say that i was about to derive the momentum operator, which is done by doing the steps backwards. I.e I demand translations symmetry and that it fulfills certain commutator relations.

It is quite simple.