PHY 742 Quantum Mechanics II

MWF 12-12:50 PM Olin 103 Webpage: http://www.wfu.edu/~natalie/s22phy742/

Instructor: Natalie Holzwarth Office:300 OPL e-mail:natalie@wfu.edu

Course schedule for Spring 2022

(Preliminary schedule -- subject to frequent adjustment.)
Lecture date
Reading
Topic
HW
Due date
1 Mon: 01/10/2022 Chap. 12 Approximate solutions for stationary states -- The variational approach #1 01/14/2022
2 Wed: 01/12/2022 Chap. 12 C Approximate solutions for stationary states -- Perturbation theory #2 01/19/2022
3 Fri: 01/14/2022 Chap. 12 D Approximate solutions for stationary states -- Degenerate perturbation theory #3 01/21/2022
Mon: 01/17/2022 MLK Holiday -- no class
4 Wed: 01/19/2022 Chap. 12 C & D Approximate solutions for stationary states -- Additional tricks #4 01/24/2022
5 Fri: 01/21/2022 Chap. 13 Examples of of the use of perturbation theory #5 01/26/2022
6 Mon: 01/24/2022 Chap. 13 & 12 B Hyperfine perturbation and also the WKB approximation #6 01/28/2022
7 Wed: 01/26/2022 Chap. 14 Scattering theory
8 Fri: 01/28/2022 Chap. 14 Scattering theory #7 02/04/2022
9 Mon: 01/31/2022 Chap. 14 Scattering theory #8 02/07/2022
Wed: 02/02/2022 No class Fire caution
Fri: 02/04/2022 No class Fire caution
10 Mon: 02/07/2022 Chap. 11 (A-C) Time evolution and Feynman path integrals #9 02/09/2022
11 Wed: 02/09/2022 Chap. 11 (A-C) Time evolution and Feynman path integrals #10 02/11/2022
12 Fri: 02/11/2022 Chap. 15 A Approximation methods for time evolution of quantum systems #11 02/14/2022
13 Mon: 02/14/2022 Chap. 15 Approximate time evolution #12 02/16/2022
14 Wed: 02/16/2022 Chap. 15 Fermi Golden Rule #13 02/18/2022
15 Fri: 02/18/2022 Chap. 15 Matrix elements and selection rules
Mon: 02/21/2022 Chaps. (11-15) Homework review & presentations
16 Wed: 02/23/2022 Chap. 16 The Dirac equation #14 02/25/2022
17 Fri: 02/25/2022 Chap. 16 The Dirac equation #15 02/28/2022
18 Mon: 02/28/2022 Chap. 16 The Dirac equation #16 03/02/2022
19 Wed: 03/02/2022 Chap. 16 The Dirac equation
20 Fri: 03/04/2022 Chap. 16 The Dirac equation
Mon: 03/07/2022 No class Spring Break
Wed: 03/09/2022 No class Spring Break
Fri: 03/11/2022 No class Spring Break
Mon: 03/14/2022 No class APS March Meeting Prepare Project
Wed: 03/16/2022 No class APS March Meeting Prepare Project
Fri: 03/18/2022 No class APS March Meeting Prepare Project
Mon: 03/21/2022 Project presentations I
Wed: 03/23/2022 Project presentations II
21 Fri: 03/25/2022 Chap. 5 & 17 Quantization of the Electromagnetic Field #17 03/28/2022
22 Mon: 03/28/2022 Chap. 17 Quantization of the Electromagnetic Field #18 03/30/2022
23 Wed: 03/30/2022 Chap. 17 Quantization of the Electromagnetic Field #19 04/01/2022
24 Fri: 04/01/2022 Chap. 18 Absorption and emission of photons
25 Mon: 04/04/2022 Chap. 10 (review) Multiparticle systems and second quantization #20 04/06/2022
26 Wed: 04/06/2022 Chap. 10 (review) Multiparticle systems and second quantization
27 Fri: 04/08/2022 Multi electron atoms #21 04/11/2022
28 Mon: 04/11/2022 Multi electron atoms #22 04/18/2022
29 Wed: 04/13/2022 Multi electron atoms
Fri: 04/15/2022 No class Holiday
30 Mon: 04/18/2022 Hubbard model with multiple electrons #23 04/22/2022
31 Wed: 04/20/2022 Hubbard model with multiple electrons
32 Fri: 04/22/2022 BCS model of superconductivity
33 Mon: 04/25/2022 BCS model of superconductivity
34 Wed: 04/27/2022 Review
35 Fri: 04/29/2022 Review

PHY 742 -- Assignment #1

January 10, 2022

Read Chapter 12, part A in Carlson's textbook.

  1. Work problem 2 at the end of chapter 12.

PHY 742 -- Assignment #2

January 12, 2022

Read Chapter 12, part C in Carlson's textbook.

  1. Work problem 6 at the end of chapter 12.

PHY 742 -- Assignment #3

January 14, 2022

Read Chapter 12, part D in Carlson's textbook.

  1. Work problem 9 at the end of chapter 12.

PHY 742 -- Assignment #4

January 19, 2022

Read Chapter 12, parts C & D in Carlson's textbook.

  1. Verify the solution and energy of the quadratic Stark effect for ground state H atom in a uniform electric field discussed in Lecture 4.

PHY 742 -- Assignment #5

January 21, 2022

Read Chapter 13 in Carlson's textbook.

  1. Derive some of the steps for estimating the Van der Waals interaction for two ground state H atoms separated by a distance a as discussed in your text book and in class.

PHY 742 -- Assignment #6

January 24, 2022

Complete reading Chapter 12 and 13 in Carlson's textbook.

  1. In class, we derived the hyperfine spliting of a hydrogen atom in its ground state using a slightly different approach than found in your textbook. Check whether the two results are compatible.

PHY 742 -- Assignment #7

January 28, 2022

Continue reading Chapter 14 in Carlson's textbook.

  1. Consider the function eix , where x is a positive quantity.
    1. Write the function as an expansion in terms of spherical bessel functions.
    2. Using Maple or Mathematica or other software, plot the real and imaginary parts of the function as a function for x in the range of 0 and 5, both for the function itself and a finite number of expansion terms.
    3. Comment on the accuracy of the expansion.

    PHY 742 -- Assignment #8

    January 31, 2022

    Continue reading Chapter 14 in Carlson's textbook.

    1. Consider the isotropic three-dimensional scattering potential V(r) which is zero for r > a and has value -V0 for 0 ≤ r ≤ a, where V0 is a positive constant.
      1. Write a general expression for the scattering phase shifts δl(E) for E >0.
      2. Evaluate the expression for l=0 for two or three values of E/V0.

    PHY 742 -- Assignment #9

    Febrary 07, 2022

    Read Chapter 11 (A-C) in Carlson's textbook.

    1. Carry out the intermediate steps to verify the result for propagator given in Eq. 11.11 of your textbook.

    PHY 742 -- Assignment #10

    Febrary 09, 2022

    Read Chapter 11 (A-C) in Carlson's textbook.

    1. Consider the time dependent wave function Ψ(xf,t) derived for the one dimensional harmonic oscillator system with mass m and frequency ω and presented on slide 15 of the lecture notes. While it is difficult to perform the integral to derive this result, it is possible to check that it makes sense in several ways. Write down the Hamiltonian for this system and work through at least one of the following.
      1. Check that the result reduces to the known form when a=0.
      2. Check that Ψ(xf,t) satisfies the time dependent Schödinger equation for the one-dimensional harmonic oscillator.

    PHY 742 -- Assignment #11

    February 11, 2022

    Start reading Chapters 15 in Professor Carlson's QM textbook..

    1. Work problem #1 at the end of Chapter 15.

    PHY 742 -- Assignment #12

    February 14, 2022

    Continue reading Chapters 15 in Professor Carlson's QM textbook..

    1. For a system with spherically symmetric states in the absence of any perturbing electromagnetic field, in the dipole approximation, there are selection rules for allowed transitions between states. What are these selection rules? Using analysis or demonstration with examples, convince yourself that these selection rules are true.

    PHY 742 -- Assignment #13

    February 16, 2022

    Finish reading Chapter 15 in Professor Carlson's QM textbook..

    1. In class we considered the transition matrix element between two eigenstates of a hydrogen like ion -- |I0=100> and |f0=210> for < f0|z|I0>. Evaluate the corresponding matrix element < f0|pz|I0>.

    PHY 742 -- Assignment #14

    February 23, 2022

    Start reading Chapter 16 in Professor Carlson's QM textbook..

    1. Write the Hamiltonian operator for a spin 1/2 free particle according to the Dirac equation in its 4 x 4 form. Show that the eigenvalues and eigenvectors are consistent with the coupled two-component form that we discussed in class.

    PHY 742 -- Assignment #15

    February 25, 2022

    Continue reading Chapter 16 in Professor Carlson's QM textbook..

    1. From the 4 × 4 matrix represenations of the operators J2 and K, show that both J2 and K2 are constant matrices and their values are related by K2=J2+ ℏ2/4. Alternatively, in terms of the quantum number J, the eigenvalues of K2 can be written K2=J(J+1)ℏ2 + ℏ2/4.

    PHY 742 -- Assignment #16

    February 28, 2022

    Continue reading Chapter 16 in Professor Carlson's QM textbook..

    1. Consider a H-like ion with Z=5 as represented by the Dirac equation. Numerically evaluate all of the eigenstate energies for principal quantum numbers n=1,2 and 3. Compare each of the results with corresponding eigenstate energies of the Schrödinger equation.

    PHY 742 -- Assignment #17

    March 25, 2022

    Read Chapters 5 and 17 in Professor Carlson's QM textbook.. In the following CQM refers to the textbook.

    1. Using Eq. 5.3 of CQM to express the displacement operator X in terms of creation and annihilation operators, show that the following expectation value is correct: ⟨n|X4|n ⟩ = (ℏ/(2mω))2(3+6n(n+1)) for the |n⟩ eigenstate of the harmonic oscillator Hamiltonian in the phonon number basis.

    PHY 742 -- Assignment #18

    March 28, 2022

    Continue reading Chapter 17 in Professor Carlson's QM textbook..

    1. Evaluate the 4 relationships between coherent states given on the last slide of Lecture 22 in order to check whether or not they are correct.

    PHY 742 -- Assignment #19

    March 30, 2022

    Finish reading Chapter 17 in Professor Carlson's QM textbook..

    1. Consider a Glauber coherent state for a single photon mode of wavevector k and polarization index σ with amplitude α=E0 exp(iψ). Calculate the average and variance of the E and B fields for this coherent state.

    PHY 742 -- Assignment #20

    April 4, 2022

    Review Chapter 10 in Professor Carlson's QM textbook..

    1. In class, we enumerated all of the distinct states for a system of distinguisable particles, Bose particles, and Fermi particles for the case where there are 2 states a and b and 2 particles. Perform the same analysis for the case where there are 3 states a, b, and c and 3 particles.

    PHY 742 -- Assignment #21

    April 8, 2022

    Review Notes for Lecture 27.

    1. Evaluate the ground state energy of a He atom using the single particle basis of the He+ ion, evaluating the expressions obtained in class. (Hint: some of these evaluations were discussed in Lecture 1.)

    PHY 742 -- Assignment #22

    April 11, 2022

    The material for this homework follows Lecture 28

    1. In class, we evaluated the ground state of a He atom and its lowest two excited states using the single particle basis of eigenstates of the He+ H-like ion with Z=2. Evaluate the same analysis for Li+ using the single particle basis of eigenstates of the Li++ H-like ion with Z=3. Compare your results with the NIST database.

    PHY 742 -- Assignment #23

    April 18, 2022

    The material for this homework follows Lecture 30

    1. Consider the two site Hubbard model, described by Hamiltonian H, with two electrons and zero total spin in the two electron basis of states |A>, |B>, and |C> discussed in the lecture. Evaluate the matrix element <A| H |C>.
    Return to main web page for PHY 712
    Last modfied: Tuesday, 15-Feb-2022 22:27:13 EST