Why can we add/subtract constants to potential function?

[MrApex]

I have 3 questions regarding the topic:
1-Why is that we are allowed to add or subtract a constant to a potential energy function V(x) to set it to zero where it is constant?
2-What does adding/subtracting a constant physically correspond to if anything at all?
3- Do we do it simply for convenience ?

Thanks in advance

 

[DrClaude]

The equations of QM (and for that matter, classical physics also) are invariant under a change of the zero of energy. All calculations will give the same result, whatever your choice of origin for energy. Therefore, it is only a matter of convenience where to set it.

For the harmonic oscillator, one usually takes the minimum of the potential to be at zero, while for the Coulomb potential in the hydrogen atom, one usually takes the zero to correspond to infinitely separated proton and electron, such that the ground state is at -13.6 eV.

 

[Sturk200]

One way to think about this is that potential energy isn't something that is directly measurable. But force is something that we can measure directly. The potential is defined as the negative spatial derivative of the force. The derivative of a constant term is zero. It follows that we can add a constant term to the potential without changing our characterization of the force. "Physically", adding or subtracting a constant corresponds to nothing at all, since it has no effect on the "physically apprehensible" force. We just pick our zero for convenience and convention.