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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.

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  • $\begingroup$ For a tensor product, the Fourier transform is simply the tensor product of the Fourier transforms on each factor. This applies generally to linear transformations and is an expression of the tensor product being functorial. $\endgroup$ Mar 20, 2015 at 17:15
  • $\begingroup$ In addition, the Fourier transform of a Gaussian is a Gaussian (easily seen completing the square in the exponential) ;-) $\endgroup$
    – yuggib
    Mar 20, 2015 at 17:21

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