Consider a particle in a 1D-Box.
The box ranges from x=0 to x=l in the first case, and from x=-l/2 to x=l/2 in the second case.
The only difference I see is that the origin is shifted.
On solving the Time Independent Schroedinger's Equation, we get different wave functions in both the cases.
Why is it that the choice of origin affects the wave function?
Do observables too depend on the choice of origin? As surely the particle "knows" that it has a region of L length to move about.