The value of 2π that we get here is mathematically a consequence of A ϕ being independent of ϕ. Physically, this means that the quantum system does not acquire a phase shift upon transportation around the z-axis, indicating that the path does not enclose non-analiticities or abrupt changes in the conditional wavefunction and that the limiting behaviour can be reached smoothly. In the case of the Z-dependence this should not surprise since our Hamiltonian is non-interacting and can therefore be trivially rescaled as Ĥ(Z, r 1 , r 2 ) = Z 2 Ĥ(Z = 1, Zr 1 , Zr 2 ).