A tricky one...

[walkerj]

What Unitary operator can transfer the ground state eigenfunction
f=1/sqrt(2*pi)*exp(-x^2/2) to the ground state eigenfunction of another harmonic oscillator f=1/sqrt(2*pi)*exp(-(x-d)^2/2)?

[slyboy]

Your new wavefunction results from taking the original system and translating it in space. Therfore, you are looking for a translation operator. I am sure you must know which quantum observable acts as the generator of translations, so you just have to exponentiate that.

Here is a clue. Generally, spacetime symmetries lead to conserved quantitites and so translation invariance leads to the conservation of .....