Adding a constant term to potential in Schrödinger's Equation If we add a constant term $k$ to the potential function in time-independent Schrödinger's equation, $V(x) \rightarrow V(x)+k$, then how does it affect the solution, and what is its significance? Especially, when it has no significance in classical mechanics.
 A: It has no significance also in quantum mechanics. The solutions  $\psi'$ of the Schrödinger equation with the new potential — viewed as an eigenvalue/eigenvector equation — are exactly the solutions $\psi$ of the Schrödinger equation with the old potential (obviously replacing $E$ for $E+k$). They are simply multiplied by a phase depending on $k$ and $t$ if we are dealing with the temporal Schrödinger equation:
$$\psi'(t)= e^{-itk/\hbar} \psi(t).$$
As pure states are unit vectors up to phases, the states are unchanged.
A: It shifts the energy. In fact, it can be moved right away into the energy part of the equation. And just like in classical mechanics it has no significance.
A bit more advanced view:
in some cases this term will originate from the gap, e.g., when reducing the Dirac equation to the non-relativistic limit or when dealing with multiband crystals. However, this usually means that we are really starting with a more complex equation, and the SE equation in question is just a part of the whole.
