Problem Set #4
Note:Relevant Chapter in Mortimer is Chapter 16.
Spin Wave Functions
1. Which of the following spin wave functions are symmetric with respect to the exchange of electrons?
2. Show that the following spin function
is an eigenfunction of the total z component of spin angular momentum for a two-electron system. What is the eigenvalue?
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 [ and ], and of the spin wave functions for each electron .
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.
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
8. Write the complete term symbols for the following states
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:
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. . 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 values of J for .