What is the Fourier transform of an infinite integral of plane waves?

Question

I'm trying to evaluate this double-integral in the context of Quantum Mechanics. Consider $f(x)$ as

$$ f(x) = \int_{-\infty}^{\infty} \mathrm{exp} \left( \frac{-ipx}{\hbar} \right) dp $$

So $\hat f(p)$, the Fourier transform of $f(x)$, is

$$ \hat f(p) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \mathrm{exp} \left( \frac{-ipx}{\hbar} \right) dp \space \mathrm{exp} \left( -2\pi i p x \right) dx $$

Numerous attempts to evaluate this has failed, though my professor has asked us to do this. Perhaps the goal is misunderstood?

Answer 1

One of the formulas for Dirac's delta function asserts that

$$\delta\left(x\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}{\rm d}k$$

Therefore, your function is just

$$f(x)=\int_{-\infty}^{\infty}\mathrm{exp}\left(\frac{-ipx}{\hbar}\right){\rm d}p=2\pi\hbar\delta\left(x\right)$$

and the Fourier transform of $f(x)$ is

$$\hat{f}(p)=\int_{-\infty}^{\infty}2\pi\hbar\delta\left(x\right)\mathrm{exp}\left( -2\pi i p x \right){\rm d}x=2\pi\hbar$$