I'm struggling to find a hermitian operator whose eigenstate is a gaussian function in $|\psi(x)|^2$. How do i do this?
Just to be clear, this is in order to realistically model the wavefunction collapse when 'position' is measured, so the eigenvalue must be the mean of the gaussian
The ground state wavefunction of the Harmonic oscillator is given by a gaussian function $$\psi_0(x)=Ce^{-m\omega x^2/2\hbar}$$ The Hamiltonian of the harmonic oscillator looks like $$\mathcal{H}=\frac{P^2}{2m}+\frac{1}{2}m\omega X^2$$ Or on a position basis $$\mathcal{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac{1}{2}m\omega x^2$$
