(a) derive the fine structure formula from the relativistic correction and the spin-orbit coupling which are given below. note that j l + 1/2; treat the plus and minus sign separately. (2 points) (en)2 2mc2 1/2 4n (1) (en)2nj1) (113/4 (11/2)(l1) e1 (2) mc2 (b) draw schematically, in one transform including the fine structure. consider n = 1,2,3 (vertical axis) and / = 0,1,2 (horizontal axis). (1 point). diagram, the energy levels of hydrogen and how they (c) we apply of the formula below, that an external magnetic field of 4 t. discuss, by hydrogen atom will experience a weak zeeman effect. (2 points) use our 1 (3) _ 4πε mear3' (d) we are particular interested to find the states under the weak-field zeeman splitting, to the fine structure state with the highest quantum numbers in the system we concerning i. e. n 3 2 ). find the energy of each state and construct a diagram to show how the energies evolve slope. (3 points) bbet creases. label each line clearly and indicate its as ji1) 1)3/4 (4) qj= 2j(j1) (e) explain, using the formula below, how the ground state of hydrogen produces the famous 21-centimeter line. (2 points) 4g, h eis 3mpm2a \-3/4 +1/4 (triplet) (singlet)