Hamiltonian of Minimum Uncertainty State

[Gza]

I was curious as to the form of the hamiltonian, whose energy eigenstate in the position basis is a gaussian distribution (or minimum uncertainty state, as I've heard from somewhere.) I haven't taken quantum for a few years, and remember studying the minimum uncertainty state as a wavefunction, without reference to its hamiltonian.

 

[Gza]

Also, does the fact that it is an eigenfunction of the Fourier transform have any physical interpretation? (aside from the obvious fact that it looks the same in the position, as well as momentum basis)

 

[StatMechGuy]

A minimum uncertainty state that's also the eigenstate of an hamiltonian is a gaussian, and it corresponds to the harmonic oscillator hamiltonian.