Let’s make it simple. If I am right, your question is pointing toward something which says,
“will a wave function, that is satisfying the SE (say Ψ) will remain a perfect wave
function when multiplied by something say ‘x’, and thus making it ‘xΨ’ ”? This question is
however a complete analog...
As we already know, a moving charge particle develops magnetic field arround itself. That mean to say, inorder to develop magnetic field arround itself a charge particle have to move. Now, let's keep the particle(charge) at rest w.r.t laboratory frame and then we starts to move; as we does so...