advertisement

CHEM 342. Spring 2002. Problem Set #4. Mortimer Chapter 16. Spin Wave Functions 1. Which of the following spin wave functions are symmetric with respect to the exchange of electrons? 1 12 2 12 2 12 12 2 2 12 12 4 2 2. Show that the following spin function 2 12 12 2 is an eigenfunction of the total z component of spin angular momentum for a two-electron system. What is the eigenvalue? 3 Slater Determinant 3. Use the Slater determinant to arrive at a wave function to describe the ground state of a two-electron system such as He. Express the resulting wave function in terms of the 1s spatial wave function for each electron [ 1s 1 and 1s 2 ], and of the spin wave functions for each electron 1, 2, 1, and 2 . Angular Momentum; Russell-Saunders coupling vs. jj-coupling; Term Symbols 4. Calculate the allowed values of j for a d electron. 5. The quantum number L represents the total orbital angular momentum, and describes l-l coupling of the orbital angular momentum of two or more electrons. Determine the values of L for two d electrons. What are the corresponding letter symbols? Hint: The quantum number L may have values between the sum of the l values of the individual electrons and the absolute value of the difference of these numbers. L l1 l 2 , l1 l 2 1,..., l1 l 2 6. The quantum number S represents the total spin angular momentum, and describes the s-s coupling of the spin angular momentum of two or more electrons. Determine the values of S for two d electrons. 7. The quantum number J represents the total angular momentum, and describes the Russell-Saunders coupling between L and S. Determine the values of J for two d electrons. Hint: the allowed values of J are given by J L S , L S 1,..., L S 8. Write the complete term symbols for the following states (a) L = 4, S = 1, J = 5; (b) L = 2, S = 0, J = 2; (c) L = 0, S = 0, J = 0 9. Determine the electronic configuration for an atom with the term symbol 4S3/2. 10. List the quantum numbers L, S, and J for the following terms symbols: (a) 4G5/2 1 CHEM 342. Spring 2002. Problem Set #4. Mortimer Chapter 16. (b) 3P2 (c) 2D3/2. 11. Derive the ground state term symbol for the following configuration (5s)1(4d)4, if given that J = 1/2. 12. Give the term symbol for Li:1s22s1. 13. Find the total angular momentum states (L, ML) for two electrons, one p-type and one d-type. 14. When spin-orbit coupling is large, Russell-Saunders coupling fails and jj-coupling must be used. It involves first coupling the individual spin and orbital momenta of the electrons into individual j values. Then, the quantum number J may have values between the sum of the j values of the individual electrons and the absolute value of the difference of these numbers. J j1 j 2 , j1 j 2 1,..., j1 j 2 . If there are more than 2 electrons, then j1 is first coupled to j2 to find a J12. Then the total angular momentum J is found by coupling j3 to J12. Find all possible 1 1 values of J for j1 1, j 2 , j 3 . 2 2 PLEASE NOTE The work you hand in should be neat and well organized, and it should show the strategy and steps you used in solving the problems, as well as the bottom-line answers (or solutions). In grading the problems, both your work-up and your final answers/solutions will be examined and evaluated. The work handed in for grading must carry a pledge that the work is entirely yours and was done without any collaboration with other persons (except for the course instructor and TA's). You are encouraged to work with others in doing the exercises and problems found in the textbook, but all work handed in for grading should be done independently. 2