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18. 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 \begin{equation} \int_{-\infty}^{\infty} \alpha |g(\alpha)|^2 d\alpha = \frac{1}{2\pi i} \int_{-\infty}^{\infty} f(x) \frac{d}{dx} f(x) dx. \end{equation} Comment: This result is of interest in quantum mechanics where it would read, in the notation of Problem 12.35: \begin{equation} \int_{-\infty}^{\infty} p |\phi(p)|^2 dp = \int_{-\infty}^{\infty} \psi^(x) \left(\frac{-ih}{2\pi} \frac{d}{dx}\right) \psi(x) dx. \end{equation*}

          18. 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 \begin{equation*} \int_{-\infty}^{\infty} \alpha |g(\alpha)|^2 d\alpha = \frac{1}{2\pi i} \int_{-\infty}^{\infty} f(x) \frac{d}{dx} f(x) dx. \end{equation*} Comment: This result is of interest in quantum mechanics where it would read, in the notation of Problem 12.35:
\begin{equation*} \int_{-\infty}^{\infty} p |\phi(p)|^2 dp = \int_{-\infty}^{\infty} \psi^*(x) \left(\frac{-ih}{2\pi} \frac{d}{dx}\right) \psi(x) dx. \end{equation*}

18. Let f(x) and g(α) be a pair of Fourier transforms. Show that df/dx and iα g(α) 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
∫-∞^∞α |g(α)|^2 dα = (1)/(2π i)∫-∞^∞ f(x) (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:

∫-∞^∞ p |ϕ(p)|^2 dp = ∫-∞^∞ψ^*(x) ((-ih)/(2π)(d)/(dx)) ψ(x) dx.

Added by Silvia S.

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