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

1. Aug 13, 2011

### Juqon

1. The problem statement, all variables and given/known data
In the derivation of the momentum operator: Why is this equal?

2. Relevant equations
[PLAIN]http://img268.imageshack.us/img268/4575/momentumoperatorderivat.png [Broken]

3. 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.

Last edited by a moderator: May 5, 2017
2. Aug 13, 2011

### vela

Staff Emeritus
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.

3. Aug 13, 2011

### Juqon

Now I see it. Thanks!