Problem Set #4 Answers
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?
The results of the permutator operator on the above wave functions are as follows
The wave functions are symmetric because the eigenvalue of the permutator operator is +1. The wave function is antisymmetric because the eigenvalue of the permutator operator is .
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?
For a two-electron system, the total z component is given by
where the spin operators and wave functions are related by
is an eigenfunction of with an eigenvalue of
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.
For a d electron, l = 2, s = 1/2. Therefore,
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.
The value of l for d electrons is 2. Therefore,
The value of L determines the letter symbol as follows, L = 0 for S; L = 1 for P; L = 2 for D; L = 3 for F; L = 4 for G; etc.
Therefore, the letter symbols corresponding to L = 4, 3, 2, 1, 0 are G, F, D, P, and S respectively.
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.
For two electrons
7. The quantum number J represents the total angular momentum, and describes the Russel-Saunders coupling between L and S. Determine the values of J for two d electrons. Hint: the allowed values of J are given by
L = 4, S = 0, J = 4
L = 4, S = 1, J = 5, 4, 3
L = 3, S = 0, J = 3
L = 3, S = 1, J = 4, 3, 2
L = 2, S = 0, J = 2
L = 2, S = 1, J = 3, 2, 1
L = 1, S = 0, J = 1
L = 1, S = 1, J = 2, 1, 0
L = 0, S = 0, J = 0
L = 0, S = 1, J = 0, 1
8. Write the complete term symbols for the following states
The format for the term symbols is . The value of L determines the letter symbol as follows, L = 0 for S; L = 1 for P; L = 2 for D; L = 3 for F; L = 4 for G; etc. This leads to the following term symbols:
9. Determine the electronic configuration for an atom with the term symbol 4S3/2.
The value of the quantum number S can be determined from the multiplicity.
This implies that there are three unpaired electrons. The S letter symbol gives . This indicates that there is one electron in each of the p orbitals because . The configuration is (1s)2(2s)2(2px)1(2py)1(2pz)1
10. List the quantum numbers L, S, and J for the following terms symbols:
The format for the term symbols is , where S is the spin quantum number,
(2S + 1) is the multiplicity, and J is the total angular momentum quantum number. The value of L determines the letter symbol X as follows, L = 0 for S; L = 1 for P; L = 2 for D; L = 3 for F; L = 4 for G; etc.
11. Derive the ground state term symbol for the following configuration (5s)1(4d)4, if given that J = 1/2.
The term symbol is 6D1/2. Note that all 5 electrons are unpaired.
12. Give the term symbol for Li:1s22s1.
so the term is 2S1/2. Note that we are only considering the unpaired electron.
13. Find the total angular momentum states (L, ML) for two electrons, one p-type and one d-type.
The quantum number L may have values between 2 + 1 = 3 and 2 - 1 = 1.
L = 3, ML = 3, 2, 1, 0, -1, -2, -3 (7 states)
L = 2, ML = 2, 1, 0, -1, -2 (5 states)
L = 1, ML = 1, 0, -1 (3 states)
The total number of states is 7 + 5 + 3 = 15.
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 .
First, add j1 and j2 to obtain . Then add to the state to obtain ; and add to the state to obtain . Therefore, . Note that there are two states with J = 1.