Express for Operator of coordinate in momentum representation

[Ed Quanta]

Homework Statement

1. Obtain an expression for operator of coordinate in momentum representation. To this end
begin with definition of the average coordinate
x = ∫ψ*(x)xψ(x)dx
express the wave functions as wave packets in terms of plane waves, and rewrite the
expression for average coordinate in such a way that it would have a form of an operator
acting on the wave functions in the momentum representation.

Homework Equations

Not sure

The Attempt at a Solution

Where do I start? I'm not sure what the question is even asking.

 

[Dick]

What kind of an operator O(p) might you apply to a plane wave exp(i*p*x) to create an expectation value integral like you have written?

 

[Ed Quanta]

I would apply an x operator. I don't see the point though.

 

[Dick]

Exactly. But you want to express the x operator in terms of p. Here's big hint. How do you express the p operator in terms of x? The point is that the same problem can be expressed in different equivalent variables. And problems that can be hard in one variable are easy in another. This is practice. That's the point.

 

[Ed Quanta]

So I just use the fact that <p>=d<x>/dt and that is the whole problem?

 

[Ed Quanta]

How can you compute <x> and <p> as functions of time?

 

[Dick]

You didn't answer my 'hint' about what the operator p looks like in the x-representation. If you want to show x and p don't commute, how do you represent p?

 

[Ed Quanta]

well we would represent p by (h bar/i)*d/dx to show x and p do not commute

 

[Dick]

Great! That's p in x-representation. Now take a plane wave and apply that and see how it 'pulls down' p. Then think of a way to 'pull down' x using an operator written in terms of p.

 

[Ed Quanta]

x = ∫(p/h bar*i) dp ?

I am not sure how to represent x in terms of p

 

[Dick]

What is the form of a plane wave solution?

 

[Ed Quanta]

Isn't the plane wave of the form exp(i(kx- wt))?

 

[Dick]

Yes. Now apply the p operator you sent me (the d/dx one).

 

[Ed Quanta]

so i get hk*exp(i(kx-wt)) and hk = p by deBroglie relations,correct?

 

[Dick]

Right again! Now in the same way define an operator using p that will make x appear in front of the plane wave.

 

[Ed Quanta]

The thing is that p is not in the exponent expression. I could pull out x using the operator -i*d/dk and d/dk=h*d/dp. But then I would have to express k in the exponent as k=p/h.

 

[Dick]

I don't see why that should pose a problem?

 

[Ed Quanta]

I think I got it now. Thanks. Muchly appreciated.

 

[babamarysol]

http://farside.ph.utexas.edu/teaching/qm/lectures/node25.html