An electron moving with non-relativistic velocity v in an electric field e experiences a magnetic fieldb given by: v x (-v(r)) v x e b=- where (r) is the electric potential. this magnetic field interacts with the magnetic moment u of the electron given by -s, =n me where s is the electron spin. assuming non-relativistic mechanics, show that the hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) s. l δη [6] r dr mec2 because of thomas precession, ah is divided by a factor of two. taking only the effect of the spin-orbit coupling into account in the hydrogen atom, calculate the energy splitting of the n = 2 excited state and the n = 3 excited state assuming that the electron could be in any of the allowed eigenstates of orbital angular momentum. draw a diagram clearly showing the splittings and label the energy levels using spectroscopic notation