PHY 742 Quantum Mechanics II

MWF 1-1:50 PM OPL 103 http://www.wfu.edu/~natalie/s20phy742/

Instructor: Natalie Holzwarth Phone:758-5510Office:300 OPL e-mail:natalie@wfu.edu

Course schedule for Spring 2020

(Preliminary schedule -- subject to frequent adjustment.)
Lecture date
Reading
Topic
HW
Due date
1 Mon: 01/13/2020 Chap. 9 Quantum mechanics of electromagnetic forces #1 01/22/2020
2 Wed: 01/15/2020 Chap. 9 Quantum mechanics of particle in electrostatic field #2 01/24/2020
3 Fri: 01/17/2020 Chap. 9 Quantum mechanics of particle in magnetostatic field #3 01/27/2020
Mon: 01/20/2020 No class Martin Luther King Holiday
4 Wed: 01/22/2020 Chap. 14 Scattering theory #4 01/29/2020
5 Fri: 01/24/2020 Chap. 14 Scattering theory #5 01/31/2020
6 Mon: 01/27/2020 Chap. 14 Scattering theory #6 02/03/2020
7 Wed: 01/29/2020 Chap. 12 Variational methods #7 02/05/2020
8 Fri: 01/31/2020 Chap. 12 Variational and other approximation methods #8 02/07/2020
9 Mon: 02/03/2020 Chap. 2,6 Single particle states of molecules and solids #9 02/10/2020
10 Wed: 02/05/2020 Chap. 2,6 H2+ molecular ion; Born Oppenheimer approximation #10 02/12/2020
11 Fri: 02/07/2020 Chap. 15 Time-dependent perturbations #11 02/14/2020
12 Mon: 02/10/2020 Chap. 15 Time-dependent perturbations #12 02/14/2020
13 Wed: 02/12/2020 Chap. 15 Time-dependent perturbations #13 02/17/2020
14 Fri: 02/14/2020 Chap. 16 The Dirac equation
15 Mon: 02/17/2020 Chap. 16 The Dirac equation #14 02/19/2020
16 Wed: 02/19/2020 Chap. 16 The Dirac equation #15 02/21/2020
17 Fri: 02/21/2020 Chap. 16 The Dirac equation #16 02/24/2020
18 Mon: 02/24/2020 Chap. 11C Path integral formalism
19 Wed: 02/26/2020 Chap. 11C Path integral formalism
20 Fri: 02/28/2020 Review
Mon: 03/02/2020 No class APS March Meeting Take Home Exam
Wed: 03/04/2020 No class APS March Meeting Take Home Exam
Fri: 03/06/2020 No class APS March Meeting Take Home Exam
Mon: 03/09/2020 No class Spring Break
Wed: 03/11/2020 No class Spring Break
Fri: 03/13/2020 No class Spring Break
Mon: 03/16/2020 No class Classes Cancelled
Wed: 03/18/2020 No class Classes Cancelled
Fri: 03/20/2020 No class Classes Cancelled
21 Mon: 03/23/2020 Chap. 17 Quantization of the Electromagnetic Field #17 03/25/2020
22 Wed: 03/25/2020 Chap. 17 Quantization of the Electromagnetic Field #18 03/27/2020
23 Fri: 03/27/2020 Chap. 17 Quantization of the Electromagnetic Field #19 03/30/2020
24 Mon: 03/30/2020 Chap. 18 Photons and atoms
25 Wed: 04/01/2020 Chap. 10 Multiparticle systems #20 04/03/2020
26 Fri: 04/03/2020 Chap. 10 Multiparticle systems #21 04/06/2020
27 Mon: 04/06/2020 Chap. 10 Multielectron atoms #22 04/08/2020
28 Wed: 04/08/2020 Chap. 10 Multielectron atoms
Fri: 04/10/2020 No class Good Friday
29 Mon: 04/13/2020 Chap. 10 Multielectron atoms #23 04/15/2020
30 Wed: 04/15/2020 Hartree-Fock and other formalisms #24 04/17/2020
31 Fri: 04/17/2020 Density functional theory
32 Mon: 04/20/2020 Density functional theory for atoms
33 Wed: 04/22/2020 Practial density functional theory
34 Fri: 04/24/2020 Brief discussion of BCS theory of superconductivity
35 Mon: 04/27/2020 Review
36 Wed: 04/29/2020 Review

PHY 742 -- Assignment #1

January 13, 2020

Read Chapter 9 in Professor Carlson's QM textbook..

  1. Verify the gauge transformation for the particle wavefunction shown in slide 11 of Lecture 1. Note that some useful steps for this verification are given on page 146 of the text.

PHY 742 -- Assignment #2

January 15, 2020

Continue reading Chapter 9 in Professor Carlson's QM textbook..

  1. Consider the stationary state solutions of the Schrödinger equation at energy E for a particle of mass m and charge q confined along the x axis by an infinite wall for x ≤ 0 and by a potential due to a constant electric field F along the x-axis qFx for x ≥ 0. In this case, the wavefunction must satisfy the boundary conditions ψ(x=0)=0 and ψ(x → ∞)=0.
    1. Set up the general equation to find the bound state energies E.
    2. Numerically determine the first 3 energies for a particular value of the field strength F.

PHY 742 -- Assignment #3

January 17, 2020

Continue reading Chapter 9 in Professor Carlson's QM textbook..

  1. Consider the full Hamiltonian given in slide 15 of Lecture 3. Assume that the magnetic field is in the z direction and is constant in time and space.
    1. Determine the second order term H2. (Note, you may want to also consult page 146 of Professor Carlson's textbook.)
    2. Consider a H atom in its ground state |nlmms>=|100 1/2>. Find the leading order perturbation theory contribution from H1+H2 for this case.

    PHY 742 -- Assignment #4

    January 22, 2020

    Start reading Chapter 14 in Professor Carlson's QM textbook..

    1. Present the steps need to show the last equation on slide 28 of Lecture 4, relating the phase shift δl to the logaritmic derivative of the radial wavefunction and the spherical Bessel and Neumann functions.

    PHY 742 -- Assignment #5

    January 24, 2020

    Continue reading Chapter 14 in Professor Carlson's QM textbook..

    1. Provide the intermediate steps to determine the asymptotic form of the scattering wave function RElscatt(r) for r → ∞ discussed on slide #14 of Lecture 5.

    PHY 742 -- Assignment #6

    January 27, 2020

    Continue reading Chapter 14 in Professor Carlson's QM textbook..

    1. In Lecture 6, slide 9, we determined the radial wavefunction PE0(r) for r ≤ a. Find the corresponding form for r ≥ a.

    PHY 742 -- Assignment #7

    January 29, 2020

    Start reading Chapter 12 in Professor Carlson's QM textbook..

    1. Solve Problem 2 at the end of chapter 12.

    PHY 742 -- Assignment #8

    January 31, 2020

    Continue reading Chapter 12 in Professor Carlson's QM textbook..

    1. Verify the variational analysis of the ground state of the He atom presented in Lecture 8, especially the Coulomb repulsion term. Extra points offered for corrected errors.

    PHY 742 -- Assignment #9

    February 3, 2020

    Review the material in Chapters 2 and 6 in Professor Carlson's QM textbook..

    1. In Lecture 9, we discussed the bound state solutions for a particle in a one-dimensional well. For the v=384 example, we found 7 bound states. Determine the energy eigenstates for this case, in terms of the dimensionless parameter ε.

    PHY 742 -- Assignment #10

    February 5, 2020

    Review the material in Chapters 2 and 6 in Professor Carlson's QM textbook..

    1. Evaluate the parameter Δ defined in Lecture 10 representing the overlap between two ψ(r)1s(r) wavefunctions separated by the distance D a0.

    PHY 742 -- Assignment #11

    February 7, 2020

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

    1. Work problem #1 at the end of Chapter 15 (page 276).

    PHY 742 -- Assignment #12

    February 10, 2020

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

    1. Complete HW #11 in PHY 712. link

    PHY 742 -- Assignment #14

    February 17, 2020

    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 #15

    February 19, 2020

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

    1. For H representing the Dirac Hamiltonian for a H-like ion and K representing the 4 × 4 "K" operator, work out the details of the commutator relationship for [H,K].

    PHY 742 -- Assignment #16

    February 21, 2020

    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 23, 2020

    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 25, 2020

    Read Chapter 17 in Professor Carlson's QM textbook..

    1. Evaluate the 4 relationships between coherent states given on page 14 of Lecture 22.

    PHY 742 -- Assignment #19

    March 27, 2020

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

    1. Verify the results presented on page 14 of Lecture 23.

    PHY 742 -- Assignment #20

    April 01, 2020

    Start reading Chapter 10 in Professor Carlson's QM textbook..

    1. Work problem #3 at the end of Chapter 10 of the textbook.

    PHY 742 -- Assignment #21

    April 03, 2020

    Continue reading Chapter 10 in Professor Carlson's QM textbook..

    1. Use the Fermi particle anticommutation relations to verify the relationships and results of Slide 12 of Lecture 26.

    PHY 742 -- Assignment #22

    April 06, 2020

    Read Chapter 10.F in Professor Carlson's QM textbook..

    1. Evaluate the expectation value of the ground state of the He Hamiltonian using the single particle basis set as used in Lecture 27. How is this result different from the result that you derived earlier in the course using perturbation theory?

    PHY 742 -- Assignment #23

    April 13, 2020

    The material for this homework follows Lecture 28

    1. Evaluate the expectation value of the ground states of the of a neutral Li atom and of a Li+ ion using single particle basis set of the H-like ion (Li++) with Z=3. From these results you can calculate the energy to ionize a Li atom and compare it with the experimental value.

    PHY 742 -- Assignment #24

    April 14, 2020

    The material for this homework follows Lecture 30

    1. Give some of the steps used to derive the Kohn-Sham and the Slater expressions for the effective exchange potentials given on slide 20 of Lecture 30.
    Return to main web page for PHY 712
    Last modfied: Thursday, 23-Jan-2020 11:18:39 EST