Question

space translation invariance in quantum mechanics
if \psi is the wave function of a 1D system is it equivalent to say that translation invariance (symmetry under translations) is fulfilled if:
a) \psi'(x+D)=\psi(x)
and
b) \psi and \psi' fulfil the same Schroedinger equation? (Which means that H=H')
I use ' to denote the quantities related to the translated system.

Answer 1

That seems right, unless there's a trick I'm missing

Answer 2

How could I show explicitly the equivalence between a) and b)?