3. Consider a system that is initially in the state: (0, 0) = = 2 −¹(0, 0) + ±³×º (0, 0) + ½±×¥³¹(0, 0), where the Y are spherical harmonics. 4 (a) If I is measured once, what possible values could be obtained and with what probabilities? Explain briefly why there are only three components of the wave function (0, 0). (b) The helium atom has two electrons. If one is in the 2p state and the other is in the 3d state, what possible values are there for the total orbital angular momentum, L, and spin, S, for the two electron system? (c) Assuming that LS coupling applies, give the term symbols (25+1 LJ) for all the possible combinations of L and S. Your answers should include all possible values of the total electronic angular momentum, J. (d) Briefly explain the origin of the most significant interaction that causes the energies of the terms, given in part (c), to be different from one an- other. What is the next most significant interaction that would need to be considered in determining the energies of the allowed states? (e) Assuming that the least significant of the two interactions relevant to part (d) is negligibly small, what is the degeneracy of the term from part (c) with the lowest value of J?

3. Consider a system that is initially in the state: (0, 0) = = 2 −¹(0, 0) + ±³×º (0, 0) + ½±×¥³¹(0, 0), where the Y are spherical harmonics. 4 (a) If I is measured once, what possible values could be obtained and with what probabilities? Explain briefly why there are only three components of the wave function (0, 0). (b) The helium atom has two electrons. If one is in the 2p state and the other is in the 3d state, what possible values are there for the total orbital angular momentum, L, and spin, S, for the two electron system? (c) Assuming that LS coupling applies, give the term symbols (25+1 LJ) for all the possible combinations of L and S. Your answers should include all possible values of the total electronic angular momentum, J. (d) Briefly explain the origin of the most significant interaction that causes the energies of the terms, given in part (c), to be different from one an- other. What is the next most significant interaction that would need to be considered in determining the energies of the allowed states? (e) Assuming that the least significant of the two interactions relevant to part (d) is negligibly small, what is the degeneracy of the term from part (c) with the lowest value of J?