Well, most intuitively, I would assume that shifting the initial wavefunction along the x-axis would affect average position.
Less intuitively, a shifted initial wavefunction in a zero-centered harmonic potential MUST be composed of eigenfunctions... ones centered at zero. Each of these have...
I have a wave function which is the ground state of a harmonic oscillator (potential centered at x=0)... but shifted by a constant along the position axis (ie. (x-b) instead of x in the exponential).
How does this decompose into eigenfunctions?? I know it's an infinite sum... but I can't...