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Homework Statement
How do you get from (3.171) to (3.172)? In particular, why is
##\int e^{-ip.r/{\hbar}}\frac{p_{op}^2}{2m}\Psi(r,t)\,dr=\int\frac{p_{op}^2}{2m}[e^{-ip.r/{\hbar}}\Psi(r,t)]\,dr##? ##\,\,\,\,\,##-- (1)
Homework Equations
The Attempt at a Solution
For (1) to be true, it must be true that
##\int\frac{\partial ^2}{\partial x^2}[e^{-ip.x/{\hbar}}\Psi(r,t)]\,dr=\int e^{-ip.x/{\hbar}}\frac{\partial ^2}{\partial x^2}\Psi(r,t)\,dr##. ##\,\,\,\,\,##-- (2)
Using integration by parts and the fact that ##\Psi(r,t)=0## at infinity, I am able to show that (2) is true provided
##\int\frac{-ip}{\hbar}e^{-ipx/\hbar}\frac{\partial}{\partial x}\Psi(r,t)\,dr=0## ##\,\,\,\,\,##-- (3) or
##\int\Psi(r,t)\,(\frac{-ip}{\hbar})^2e^{-ipx/\hbar}\,dr=0##.##\,\,\,\,\,##-- (4)
But I don't know how (3) or (4) is true?