First some context to my question:
I have two sets $M=\{p(x)e^{-x^2}:p\in \mathbb{C}[x]\}$ and $N=\{\hat{f}:f\in M\}$. Both are left modules of the Weyl algebra $A_1$. There are a few other technical details that I will not get into because they would not be relevant to my question. I need to show that $M\cong N$ as $A_1$-modules. I have defined a mapping from $M$ to $N$ as $f \mapsto \hat{f}$. I am trying to prove that this mapping is a bijective $A_1$-module homomorphism.
To show that I have an $A_1$-module homomorphism, I first need to know what the Fourier transform of the type of functions in $M$ looks like. It used to be that I knew how to calculate the Fourier transformation, but that was some time ago. I would appreciate if someone can provide an answer to this question or perhaps point to an article that contains the answer. I am not that interested in the details of how to find the answer; just knowing what $\hat{f}$ is suffices for my purposes.
The definition of $\hat{f}$ in the notes I am reading is
If it makes no differene to you, I'd use $e^{-\frac{1}{2}x^2}$ instead, which simplifies the formulas. Then you can use the fact that $h_n(x) = (-1)^n\frac{d^n}{dx^n} e^{-\frac{1}{2}x^2}$ is of the form $p_n(x)e^{-\frac{1}{2}x^2}$ where $p_n$ is a monic polynomial of degree $n$. (cf. Hermite polynomials, Hermite functions) In particular, you can write any polynomial $p$ as a linear combination of such polynomials:
$$ p(x) e^{-\frac{1}{2}x^2} = \sum_{k=0}^{\deg p} \alpha_k h_k(x) $$
And $h_n$ is trivial to Fourier transform, since $\mathcal F[\frac{d^n}{dx^n} f](w) = (iw)^n \hat f(w)$. In particular, the Fourier Transforms are again functions of the same form. This is because the Hermite functions are eigenfunctions of the Fourier Transform. If you stick with $e^{-x^2}$ then nothing substaintially changes besides that you will get the exponent $e^{-w^2/4}$ in the ferquency domain (https://en.wikipedia.org/wiki/Fourier_transform#Square-integrable_functions,_one-dimensional)