Fourier Transform of probability density in real space as a convolution of its frequency amplitudes

Question

I'm trying to find an efficient method to calculate the Fourier transform of a probability density at a cylindrical boundary an infinite distance away given some arbitrarily placed point emitters within a finite displacement from the center.

Effectively the un-normalized positional probability amplitude at some angle from a reference point along the cylinder is: $$\psi (\theta) = \sum_j \exp(i \nu r_j \sin(\theta - \alpha_j - \pi/2)) $$

The coordinate space is polar, each emitter $j$ is located at radius $r_j$ at angle $\alpha_j$.

The momentum probability amplitude, the FT of the positional probability amplitude becomes: $$ \phi (k) = \sqrt{2 \pi} \sum_j \exp(-i k (\alpha_j+\pi/2)) J_k(\nu r_j)$$ from the expression $$ \phi(k) = \frac{1}{\sqrt{2 \pi}} \int \exp(-i k \theta) \psi(\theta) $$

$J_k$ is the Bessel function of the first kind of order $k$ , Where $k$ can only be an integer and $ \phi(k)$ rapidly approaches $0$ as $|k|\gg 0$, this is relative to the scaling factor $\nu$ and the corresponding $r_j$.

I verified Parseval's theorem for these expressions such that

$$ \int^{\pi}_{-\pi} \psi^{\dagger} (\theta) \psi(\theta) d \theta = \sum_{m=-q}^{q} \phi^{\dagger}(k) \phi(k)$$ where $q$ is the largest frequency with a non-zero coefficient.

So in my simulation, I have for numerical/practical purposes, about 10 non-zero components for the entire space for $ \phi(k) $. So calculating the wave function from $ \phi(k) $ is much easier than using $\psi(\theta)$ because there could be hundreds of point sources and depending on my choice of $\nu$, fewer non-trivial frequencies.

So I recall from the convolution theorem that $$\psi^{\dagger}(\theta)\psi(\theta) = \phi^{\dagger}(k) \ast \phi(k) $$ All of the convolved Fourier components become real. That doesn't make sense to me because it means the $\alpha_j$ components vanished and the information in the function got corrupted.

To give you a reference of what I'm trying to do: I basically have a bunch of sets of point distributions that hint at shapes (e.g. elliptical, boxy, triangular clusters) and I want to see if I can use a wave function to classify various clusters of points according to their ensemble shape without thinking about what a shape means.

I'm going to keep looking over my code to see if I botched the convolution, but I'm hoping someone here can see what I wrote and maybe find a flaw in my logic so I can see if this idea works.

Thanks!

EDIT: I added the phase $\pi/2$ to match my code, it made me realize that I think I have a phase problem in the expression. I originally had rotated the original phase by $\pi/2$. The convolution expression changes values if I remove the constant offset, though all the phases still cancel out.

Answer 1

I found my error.

$$\int \psi^{\dagger}(\theta) \psi(\theta) d \theta = \sum_k \phi^{\dagger} (k) \phi (k)$$ is always true.

However, that does not imply the following relation I mentioned. In general: $$\psi^{\dagger}(\theta) \psi(\theta) \neq \mathscr{F^{-1}} [\phi^{\dagger}(k) \ast \phi(k)] $$ More specifically, my precise error was that in general $$\phi^{\dagger}(k) \neq \mathscr{F}[\psi^{\dagger}{\theta}] $$

In fact, what I wanted was: $$\psi^{\dagger}(\theta)\psi(\theta) = \mathscr{F^{-1}} [\mathscr{F}[\psi^{\dagger}(\theta)] \ast \mathscr{F}[\psi(\theta)]]$$

What I had before in my question was a reflection of the phases (the complex conjugate in momentum space), which caused all the angle information to cancel out in the subsequent convolution. I wanted a rotation in the phases due to the reflection in real space because I was interested in the components of the positional probability density. Now things are looking better and I can continue refining my code. This topic can be closed now.