518 (W10) Daily lecture topics

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This page lists what was covered in lectures, reading assignments, and also archives handouts. GY refers to Gottfried and Yan.

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Lecture Date Covered in lecture Reading (not covered or partially covered)
1.1 1/3 Class organization.
Mixed ensembles and density operator (Sakurai 3.4).
( Notes )
You should read Sakurai's discussion of quantum statistical mechanics---we will not cover it in class.
Further reading is GY 2.2.  
1.2 1/5 Time evolution of density operator
Mixed states as subsystems of pure states
Begin discussion of rotations and group SO(3)
( First pretest )
( Notes )
Sakurai 3.1. I am covering more background on group theory than in Sakurai, so you may want to look at some of the group theory texts listed here .  
1.3 1/7 Finish discussion of SO(3): generators and their commutation relations
Rotation operators in QM, using ang. mom. operators as generators
Deriving ang. mom. commutation relations.
( Notes )
2.1 1/11 Vector operators (Sakurai 3.10)
Rotations acting on spin-1/2 particle (Sakurai 3.2)
( Notes )
2.2 1/13 Experimental tests of spin-1/2 rotation operators (Sak. 3.2)
Why did we end up with SU(2) and not SO(3)?
SU(2) versus SO(3) (Sak 3.3)
Begin representations of angular momentum (Sak 3.5) ( Notes )
Read about details of the neutron interferometry experiment in Sak. 3.2
SU(2) vs. SO(3) is discussed in Sakurai 3.3---in particular alternative parameterization of the group is shown  
2.3 1/14 Complete discussion of representations of ang. mom
Wigner functions and Euler rotations
Discussion of HW1
( Notes )
Optional: read Sakurai 3.8, which describes a method to calculate Wigner functions in general  
3.1 1/18 HOLIDAY!  
3.2 1/19 Can all states of given j can be rotated into one another?
Orbital angular momentum and Spherical harmonics (Sak. 3.6)
( Notes )
Read Sakurai section 3.6 for details which I skipped
You should definitely read the last section on the relation between rotation matrices and spherical harmonics
3.3 1/21 Completeness of spherical harmonics
Adding angular momenta and C-G coefficients (Sak 3.7)
( Notes )
Last half-hour: qual problem
4.1 1/24 Addition of ang. mom.--sketch of general method
Introduction to spherical tensor operators (Sak. 3.10) ( Notes )
Read "Clebsch-Gordon Coefficients and Rotation Matrices"--last section of 3.7  
4.2 1/27 Spherical tensor operators: what they are and how to construct them.
Statement of Wigner-Eckart theorem (Sak. 3.10) ( Notes )
My sketch of a proof uses finite rotations; read Sakurai 3.10 for an equivalent proof using infinitesimal transformations.  
4.3 1/29 Proof of W-E theorem
Examples (including "projection theorem") ( Notes )
Qual problem
5.1 1/31 Finish discussion of qual problem
Recap of 3-dim Schr. equations: general method ( Notes )
Sakurai assumes all this material, and gives a summary of results in App. A
Read Sakurai 4.1 for a nice summary of symmetries in QM
For a more detailed discussion of the 3-d Schr. Eq. see G+Y 3.6 and 5.1-2 (or almost any other QM text).  
5.2 2/2 Solving radial equation V=0: Sketch of Infield's method for generating spherical Bessel fcns
Bound states of spinless hydrogen atom: asymptotic forms and general ansatz. ( Notes )
5.3 2/4 Recursion relation for polynomials and quantization condition for spinless hydrogen atom.
Understanding the degeneracies using the Runge-Lenz vector. ( Notes )
Gottfried and Yan discuss the Runge-Lenz vector in sec. 5.2
6.1 2/7 Brief summary of continuous symmetries in QM (Sak. 4.1)
Brief recap of parity operator (Sak. 4.2)
Time reversal invariance (Sakurai 4.4)
Motion reversal operator, need for antiunitary operator, form for spinless particle ( Notes )
Read sakurai 4.1 and particularly 4.3: discrete translation symmetry. The latter will not be discussed in lectures.  
6.2 2/9 Continue time reversal invariance:
consequences if H is invariant: second solution, simultaneous H and T eigenvectors (although eigevalue changes with time), relations between matrix elements, transformation of spin and angular momentum, begin discussion for spin-1/2. ( Notes )
Read the section in Sakurai 4.4 on spin-1/2, which gives a different way of determining the form of the time-reversal operator, and some more applications.  
6.3 2/11 Finish up time-reversal: complete spin 1/2 and discover Kramers degeneracy. ( Notes )
Review for midterm.
Qual problem
7.1 2/14 MIDTERM.
Solution .
7.2 2/16 Begin time independent perturbation theory: non-degenerate case and example from SHO. (Sak 5.1) ( Notes )
7.3 2/18 Quadratic stark effect as example of non-degerate PT
Brillouin-Wigner PT
Begin degenerate PT (Sak 5.2) with example of linear Stark effect
( Notes )
Read those parts of Sakurai 5.1 which I do not discuss in class: the 2x2 case and Van de Waals interactions.
8.1 2/21 HOLIDAY  
8.2 2/23 Formalism of degenerate PT including second order term (Sakurai 5.2)
Application to n=2 Stark effect
Sakurai problem 5.12
( Notes )
Read Sakurai 5.2 for higher order terms in degenerate case.  
8.3 2/25 Finish discussion of Sak. prob. 5.12
The "real" hydrogen atom: fine structure. (Sakurai 5.3 has a patchy discussion---GY 5.3 has more.) ( Notes )
You should read about the Zeeman effect in 5.3
I will not discuss Variational methods since we covered them in 517.  
9.1 2/28 Pretest
Start discussion of time independent PT (5.5)
Interaction picture and Dyson series for time evolution operator
Solving 2 state problem with oscillating off diagonal perturbation: set up
( Notes )
Read Sakurai's discussion of two-state problems (in 5.5), both to see the generality of this example, and a different method of solution.  
9.2 3/2 Exact solution of 2-state problem in rotating frame
Constant V and Fermi's Golden Rule (Sak 5.6) ( Notes )
9.3 3/4 Harmonic V and extension of Fermi's Golden rule (Sak 5.6)
Application to photoelectric effect (Sak 5.7) ( Notes )
We will cover radiative transitions in 519, when we treat the photon quantum mechanically.  
10.1 3/7 Sudden approximation
Adiabatic theorem and using it to obtain time indep PT from time dep. PT
Begin discussion of Berry's phase (Sak. Supplement I and GY 7.7) ( Notes )
10.2 3/9 Complete discussion of Berry's phase
Corrections to adiabatic approximation ( Notes )
10.3 3/11 Review for final exam
Example qual questions
11 3/15 FINAL EXAM (10:30-12:20)    

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Stephen Sharpe
Last modified: Wed Mar 12 11:11:21 PDT 2008