I Problem during finding <x> in p space

[Shell_E]

Thank you advance for help!

So the problem is find <x> in P space.
The solution's logic is use formula:

The submit with

Then one will get the integral

by noticing that

, one can replace xe^ipx/ħ in the integral

so that

and then after solving the integral one can find the expectation value for x.

The part I don't understand is why

, firstly I am not sure where is the minus sign comes from. Next I don't think I can just change the order of e^ipx/ħ and Φ because the term d/dp.

Could anyone help me figure the integral out?

Thank you everybody.
Shell

 

[Jilang]

I think that integration by parts comes into this one.

 

[DrClaude]

It comes about because momentum is a Hermitian operator: http://www.colby.edu/chemistry/PChem/notes/MomentumHermitian.pdf

 

[Shell_E]

@Jilang @DrClaude Thank you for help! I mislead by the integral sign to think it gained by doing some rearrangement inside the integral. Everything become clear if I do integral by part and use the fact that momentum operator is hermitian. Thanks a lot!

 

[dextercioby]

The tacit assumption is that any wave function decreases at infinity slower than any polynomial, thus can be safely assumed to be a Schwartz test function. The Schwartz space is invariant under the Fourier transform and provides a domain of essential self-adjointness for the momentum operator over the entire real line.
Because of this explanation, you can use partial integration in the improper definite integral and discard the so-called "surface term" (a misnomer, because we're in 1D).