I would like to evaluate the Fourier Transform of $n$ functions. I am aware from the derivation of the convolution how this is done for the case of $n=2$. How could this be generalised for $n=3$?
Namely, defining the Fourier transform of a function $g(x)$ to be
$g(k) = \int^\infty_{-\infty} dx\,g(x)e^{-ikx}$,
and its inverse to be
$g(x) = \frac{1}{2\pi}\int^\infty_{-\infty} dk\,g(k)e^{ikx}$,
how could one determine the Fourier Transform of the following ket (wavefunction)
$|\psi_x\rangle = \otimes_{i=0}^n \left(\int_{-\infty}^\infty \, dx\,f(x_i)\hat{a}^\dagger(x)\right)|0\rangle$. The wavefunction describes the production of $n+1$ Guassian sources centered at positions $(x_0, x_1, \ldots, x_n)$. The subscript $x$ implies that this is the wavefunction in the spatial domain. How would one determine the Fourier transformation of this. Namely, what is
$|\psi_k\rangle = \int^\infty_{-\infty} dx_0dx_1 \cdots dx_n\,|\psi_x\rangle e^{-i\sum_jk_jx_j} = \int^\infty_{-\infty} dx_0dx_1 \cdots dx_n\,\otimes_{i=0}^n \, h(x_i) e^{-i\sum_jk_jx_j}$, where we have defined
$h(x_i) = \int dx\,f(x_i)\hat{a}^\dagger(x)$ with $f(x_i) = \frac{1}{(2\pi\sigma^2)^{1/4}}e^{\frac{(x-x_i)^2}{4\sigma^2}}$ - a Guassian wavepacket.
So in summary I would like to determine the final form of $|\psi_k\rangle$. Thanks.