- #1
Happiness
- 679
- 30
Homework Statement
Homework Equations
The Attempt at a Solution
First substitute ##\Phi(p,t)## in terms of ##\Psi(r,t)## and similarly for ##\Phi^*(p,t)##, and substitute ##p_x^n## in terms of the differentiation operator
##< p_x^n>\,=(2\pi\hbar)^{-3}\int\int e^{ip.r'/\hbar}\Psi^*(r',t)\,dr'\int(i\hbar)^n(\frac{\partial^n}{\partial x^n}e^{-ip.r/\hbar})\Psi(r,t)\,dr\,dp##
Then integrate by parts with respect to ##x##. The integrated part vanishes since ##\Psi(r,t)## equals to 0 at infinity. We have
##=(2\pi\hbar)^{-3}\int\int e^{ip.r'/\hbar}\Psi^*(r',t)\,dr'\int(i\hbar)^{n-1}(\frac{\partial^{n-1}}{\partial x^{n-1}}e^{-ip.r/\hbar})\Big[-i\hbar\frac{\partial}{\partial x}\Psi(r,t)\Big]\,dr\,dp##
Then I'm stuck because it seems like we need to integrate by parts with respect to ##x## again for another ##(n-1)## times in order to get
##=(2\pi\hbar)^{-3}\int\int e^{ip.r'/\hbar}\Psi^*(r',t)\,dr'\int e^{-ip.r/\hbar}\Big[(-i\hbar)^n\frac{\partial^n}{\partial x^n}\Psi(r,t)\Big]\,dr\,dp##.
But now the integrated part no longer vanishes since ##\frac{\partial}{\partial x}\Psi(r,t)## and higher derivatives may not be 0 at infinity. So we can't get the above that we need.
The proof for the case of ##< p_x>## is attached below for reference.