# Momentum Operator Proof

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

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siddharth
Homework Helper
Gold Member
Could you be more specific? Proof of what?

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
Homework Helper
Gold Member
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
Homework Helper
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
Gold Member
There is no *proof*, but there is a "classical" way to calculate it. But still its 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 youll get the well known result:
<P>=(Ψ,pΨ), where p is the momentums operator.

malawi_glenn
Homework Helper
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
Gold Member
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 momentums 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
Homework Helper
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.