What is the solution to [x,p2] and [x,p2]ψ(x) in quantum mechanics?

[atomicpedals]

Homework Statement

Using the results of the previous problem, find [x,p² ] and from that determine [x,p² ]\psi(x)

Homework Equations

The solution to the previous problem was [A,BC]=[A,B]C+B[A,C]

The Attempt at a Solution

As I'm suppose to use the results of the previous problem I think what I need to do is

[x,pp]=[x,p]p+p[x,p]

But that doesn't really seem right to me... shouldn't this lead me to an identity (it doesn't say but usually in a QM problem p is momentum and this would seem to be leading to a wave equation in momentum space). If I really am approaching this the correct way, then how should I treat [x,p² ]\psi(x)?

 

[George Jones]

What is [x,p]?

 

[George Jones]

Yes, you are on the right track. I should have started my previous post with this.

 

[atomicpedals]

My first instinct is to answer [x,p]=-[p,x]...but I don't think that was what you were asking?

 

[George Jones]

My first instinct is to answer [x,p]=-[p,x]...but I don't think that was what you were asking?

While this true, you need to evaluate [x,p] explicitly. The answer is something simple, and will have been covered earlier in your text/course.

 

[atomicpedals]

Am I correct in thinking that x is position and p momentum?

 

[atomicpedals]

The canonical commutation? [x,p]=i \hbar

 

[George Jones]

Yes; now, use this in the expression at the end of your first post.

 

[atomicpedals]

(i \hbar)p+p(i \hbar) ?

 

[George Jones]

Right, but remember that \left[ x , p \right] = i\hbar really means \left[ x , p \right] = i\hbar I, where I is the identity operator.

 

[atomicpedals]

So in the first part I then end up with a result of

(i \hbar I)p + p(i \hbar I)

Do the standard rules of algebra apply here allowing me to combine this?

Then for the second part is my result just the combined result with the \psi(x)? (a complex wave equation in momentum space?)

 

[George Jones]

So in the first part I then end up with a result of

(i \hbar I)p + p(i \hbar I)

Do the standard rules of algebra apply here allowing me to combine this?

Yes, i\hbar I commutes with all operators, and IA = A for all operators A.

Then for the second part is my result just the combined result with the \psi(x)? (a complex wave equation in momentum space?)

\psi \left( x \right) is a complex wave function (not equation) in position space (since \psi is a function of position x). What is the position space representation of the momentum operator, i.e., p = ? This also will have been covered earlier in your text/course.

 

[atomicpedals]

I think I'm starting to grasp this. It seems like the final solution is heading towards something similar to p_x = i \hbar \partial/\partialx .