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

In summary: You can derive the momentum operator p from the commutation relations given by p=m*d*x/dt. However, this is not a proof because you don't actually know what p equals.
  • #1
csutton1
2
0
Does anyone have a momentum operator proof? The book I'm using is skipping a lot of steps .
 
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  • #2
Could you be more specific? Proof of what?
 
  • #3
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.
 
  • #4
csutton1 said:
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.
 
  • #5
I am also stunned, are you saying that you want a proof that

[tex] -i\hbar \nabla [/tex]

gives the momentum of a state?

Be specific, and write down for us where you get stucked.
 
  • #6
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.
 
  • #7
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.
 
  • #8
malawi_glenn said:
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?
 
  • #9
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.
 

What is the momentum operator?

The momentum operator, denoted as p, is a mathematical operator used in quantum mechanics to describe the momentum of a particle. It is defined as the derivative of the position operator with respect to time.

How is the momentum operator derived?

The momentum operator is derived using the principles of quantum mechanics, specifically the wave-particle duality. It is obtained by applying the de Broglie relation (p = h/λ) to the Schrödinger equation, which describes the behavior of quantum particles.

What is the physical significance of the momentum operator?

The momentum operator is used to determine the momentum of a particle in a given quantum state. It is a fundamental quantity in quantum mechanics as it is related to the uncertainty principle, which states that the more precisely the momentum of a particle is known, the less precisely its position can be known.

How is the momentum operator represented mathematically?

The momentum operator is represented mathematically as a differential operator. In one-dimensional space, it is written as p = -iħ(d/dx), where i is the imaginary unit and ħ is the reduced Planck's constant. In three-dimensional space, it is written as p = (-iħ∇), where ∇ is the nabla operator.

How is the momentum operator used in quantum mechanics?

The momentum operator is used in quantum mechanics to calculate the expectation value of the momentum of a particle in a given state. It is also used to determine the time evolution of a quantum system and to solve problems involving momentum in quantum mechanics.

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