Problem 1 (15 points) Adding angular momenta. As vectors, angular momenta and spin don't add like regular numbers, and this has many significant effects in atomic and nuclear physics, and also in elementary particle physics. For example, when bound states are formed by combining two spin1/2 particles into a bound state, one can get one kind of particle with spin 0 and another kind of particle with spin 1. The point of this exercise is to make you familiar with the idea so that you know it when we see it in applications. The rule for addition of any two angular momenta J1 and J2 (they can be spin or orbital angular momenta or total angular momenta) is that J can take any value that is an integer difference between J1 + J2 and |Jl 7 .ng (i.e. Jl + J2.Jl + .12 7 1, .11 + Jg 7 2, lJl 7 ng) Suppose you have two particles with orbital angular momentum quantum numbers l1 : l2 : 1 and spin quantum numbers 51 2 5'2 2 1/2 (a) What are the possible values of the quantum number a for the total spin S = 81 + 32? What are the corresponding magnitudes of S? (b) What are the possible values of the quantum number 5 for the total orbital angular momentum L 2 L1 + L2? What are the corresponding magnitudes of L\"? (c) Use the results of (a) and (b) to find the possible values of the quantum number j for the total angular momentum J : L + S? What are the corresponding magnitudes of J? Problem 2 (15 points) Spin-orbit effect This problem illustrates a physical implication ofthe spinorbit effect. The spectrum of sodium (Na, Z : 11) displays a prominent yellow doublet line arising from transitions from the 2103/2 and 2191/2 states to the ground state 13. The wavelengths of these two lines are 58911111 and 589611111. The reason for the lifting of the 2P degeneracy is the spinorbit effect. (a) Calculate the energies Egpa and Egg/2 (in eV) of the photons with these wavelengths. b lnfer from ou answers in a what is the ener splittin AE of the 2103 2 and 2P1 2 states. y gy g / / (c) Calculate the relative energy splitting: AE/Eggm. Note that the energy split caused by spinorbit effects are small compared to the usual energy differences in 2P to 18' transitions