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I have the function $\psi$ defined with the Fourier(FT) and inverse Fourier transform(IFT) between the $x$-$k$ space such that $$ \psi(x)=\mathcal{F}\{\psi_{k}\}=\frac{1}{\sqrt{L}}\int_{-\infty}^{\infty}dk\,e^{ikx}\psi_{k} \\ \psi_{k}=\mathcal{F}^{-1}\{\psi(x)\}=\frac{1}{\sqrt{L}}\int_{-\infty}^{\infty}dx\,e^{-ikx}\psi(x) $$ where $L$ is a normalization constant. How can I show, and is it true, that $\psi_{k}^{2}$ is given as $$ \psi_{k}^{2}=\mathcal{F}^{-1}\{\psi^{2}(x)\}=\int_{-\infty}^{\infty}dx\,e^{2ikx}\psi^{2}(x) $$ ? Otherwise, what should be the correct approach in writing $\psi^{2}_{k}$ in $x$-space? I understand the convolution theorem can be used here, but I'm not sure how to proceed or relate that to solving the problem.

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  • $\begingroup$ @flyinginsectleopard Thanks for your comment. So how should one evaluate $\psi_{k}^{2}$ in the fourier basis? $\endgroup$
    – kowalski
    Aug 8 at 20:37

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