Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. It only takes a minute to sign up.
Sign up to join this community
I have a diagonalization problem. I have the eigenstates correctly, and I want to do a Gaussian-smearing (Weierstrass-transform) on them. So I have the wave functions ($\Psi$), and the continuous equation:
$$\zeta(x)=\int \mathrm{d} x' g_{\sigma_x}\left(x-x'\right)\left|\Psi(x')\right|^2,$$
where $g_{\sigma_x}$ is a normal distribution.
I do not know, how to do this with discrete vectors in MATLAB, i.e., $|\Psi|^2, \zeta \in \mathbb{C}^n$.
What you want is the convolution between two functions $f = |\Psi|^2$ and $g = g_{\sigma_x}(x)$, $h = (f * g)(x)$.
You can compute the Fourier transform of $h$, to get
$$\mathcal{F}\{h\} = \mathcal{F}\lbrace f\rbrace \mathcal{F}\lbrace g\rbrace\, ,$$
and then, just compute the inverse Fourier transform to obtain what you want
$$ h = \mathcal{F}^{-1}\lbrace\mathcal{F}\lbrace f \rbrace \mathcal{F}\lbrace g\rbrace\rbrace\, .$$
To do that in MATLAB, you need to sample your functions over your domain and use the Fast Fourier Transform instead. Maybe, MATLAB already has something like fftconvolve.
