Chapter 7.13, Problem 18MP

Let $f(x)$ and $g(\alpha )$ be a pair of Fourier transforms. Show that $df/dx$ and $i\alpha g(\alpha )$ are a pair of Fourier transforms. Hint: Differentiate the first integral in (12.2) under the integral sign with respect to x. Use (12.23) to show that ${\displaystyle {\int }_{-\infty }^{\infty }\alpha }|g(\alpha ){|}^{2}d\alpha =\frac{1}{2\pi i}{\displaystyle {\int }_{-\infty }^{\infty }\bar{f}}(x)\frac{d}{dx}f(x)dx$.

Comment: This result is of interest in quantum mechanics where it would read, in the notation of Problem 12.35: ${\displaystyle {\int }_{-\infty }^{\infty }p}|\varphi (p){|}^{2}dp={\displaystyle {\int }_{-\infty }^{\infty }{\psi }^{*}}(x)\left(\frac{-ih}{2\pi }\frac{d}{dx}\right)\psi (x)dx$.