Derivation of momentum operator / origin of -ih d/dx

[Juqon]

Homework Statement

In the derivation of the momentum operator: Why is this equal?

Homework Equations

[PLAIN]http://img268.imageshack.us/img268/4575/momentumoperatorderivat.png

The Attempt at a Solution

I thought maybe the integral is executed, but the integral sign stays.
One p vanishes, the momentum operator itself shows up.
You would really expect this in a derivation of p, but dp stays.

 

[vela]

There's two steps in there. First, you can write
$$p \exp\left(\frac{ipx}{\hbar}\right) = \frac{\hbar}{i}\frac{\partial}{\partial x}\exp\left(\frac{ipx}{\hbar}\right)$$
Then you interchange the order of integration and differentiation.

 

[Juqon]

Now I see it. Thanks!