517 (Au10) Daily lecture topics

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This page lists (before the lectures) what is planned to be was covered, and then (after the lectures) what was actually covered. It also lists material that you are expected to read that was not covered, or only partially covered, in lectures. Handouts are also archived here.

To bottom (most recent lectures)

Lecture Date Covered in lecture Reading (not covered or partially covered)
1.1 9/30 Class organization.
Sakurai 1.1: Stern-Gerlach experiment and spin in QM.
( Notes )
Analogy with circular polarization in Sak. 1.1.  
1.2 10/1 This will be a regular lecture.
Recap on complex vector spaces. Dirac notation (Sak. 1.2).
First handout .
2.1 10/5 More quantum kinematics, mainly from Sakurai sections 1.2 and 1.3:
Spin bases, operators, outer products, hermitian operators, orthonomal bases. ( Notes )
You will have to read ahead to the end of 1.3 to cover all the material on HW1  
2.2 10/7 Matrix representations of operators, trace, determinant.
Measurement theory in QM (Sakurai 1.4)
( Notes )
What happens when eigenvalue measured is degenerate? What replaces the projection operator in this case? Read last page of notes (4.10)
2.3 10/8 Wrap up discussion of measurement
Compatible and incompatible observables ( Notes )
3.1 10/12 Heisenberg Uncertainty Principle (Sak. 1.4)
Unitary operators and basis changes (Sak. 1.5)
( Notes )
Read "Unitarily equivalent observables" (Sak. 1.5)
Do exercise given in class.  
3.2 10/14 Continuously labeled basis kets (non-normalizable)
Classical to quantum "transition" for a single, spinless particle (Dirac's correspondence, begin Sakurai's approch)
( Notes )
Read 48-51 in Sakurai (last four pages of section 1.6)  
3.3 10/16 Finish Sakurai's discussion of classical to quantum transition
Momentum, position and translation ops (Sak 1.6)
Momentum operator in position basis, momentum basis states (Sak 1.7)
( Notes )
4.1 10/19 Finish Sakurai 1.7: transforming between position and momentum bases and Gaussian wave packets
Begin Ch. 2: time evolution operator, Schrodinger equation, general solutions. ( Notes )
Read last part of Sakurai Ch. 1 concerning wave packets.  
4.2 10/21 Example of time evolution: precession of a quantum spin
Schrodinger vs. Heisenberg pictures ( Notes )
Read Sakurai pages 78-80: Energy-time "uncertainty" relation.  
4.3 10/22 Examples of Heisenberg equations of motion:
Free particle, Ehrenfest theorem, spin-1/2 precession
( Notes )
5.1 10/26 Wave-packet spread
Time evolution backward in time
Heisenberg picture base-kets (Sak. 2.2, pp87-89)
Begin Simple Harmonic Oscillator (Sak. 2.3) First pretest .
( Notes )
5.2 10/28 General considerations for solving time indep. Schrodinger eq:
graphical analysis, asymptotic forms, reality of wavefunctions, discrete and continuous spectra.
Parity operator
( Notes )
I am assuming that you have some familiarity with the derivation of SHO wavefunctions in terms of Hermite polynomials. Please read Sakurai A.4 to refresh your memory.
The other topics are scattered in Sakurai, or appear in any other QM text.  
5.3 10/29 Finish discussion of parity operator
Dirac's operator solution of SHO
( Notes )
6.1 11/2 Matrix elements in the SHO
Variational priniciple and application to SHO
Coherent states in the SHO
( Notes )
For more on coherent states see Gottfried and Yan, pp181-4.
6.2 11/4 Connecting our definition of coherent states with the standard one
Squeezed states in the SHO
Some odds and ends on the SHO
( Notes )
For more on squeezed states, you'll need to look beyond the standard texts. Wikipedia has some good references.  
6.3 11/5 Review for midterm  
7.1 11/9 MIDTERM
Solution is here .
7.2 11/12 1-d scattering: quick review
Solving an example scattering problem
( Notes )
You should be able to reproduced the results collected in Sakurai appendices A.1-A.3.
Sakurai has little discussion of 1-d scattering, but most quantum texts do. See for example GY sec 4.4.  
8.1 11/16 Plots of transmission probability for potential barrier, and Mathematica notebook to create them.
Interpretation of time-independent scattering solutions using probability current density
Begin discussion of scattering using wave packets
( Notes )
Probability current density is discussed on Sakurai pp101-3.
See Baym for a nice discussion of scattering using wave packets.  
8.2 11/18 Complete discussion of scattering using wave packets
Various plots shown are here (or in Mathematica notebook ).
Begin discussion of bound-state energies from poles in T and R.
( Notes )
8.3 11/19 Complete discussion of bound states
Begin discussion of WKB approximation: derivation. ( Notes )
Read Sakurai pp.103-109.  
9.1 11/25 Classes canceled!    
9.2 11/25 HOLIDAY!    
9.3 11/26 HOLIDAY!    
10.1 11/30 WKB approximation in 1-d: general solution, matching equations, application to bound state problems
Handout and notes .

Mathematica notebook
used to obtain WKB and exact wavefunctions for example of SHO with hard walls and quartic potential.
For further reading, see Gottfried and Yan, sec 4.5(b), and L.S. Brown, Am. J. Phys. 40, 371 (1972) ["Classical limit and the WKB approximation"] and 41, 526 (1973) ["Classical limit of the Hydrogen Atom"].
Notes and handout for an additional lecture (not given due to snow day) on the decay of resonances using wave-packets and the WKB approximation.
This is adapted from B.R. Holstein, Am. J. Phys. 64, 1061 (1996) "Understanding alpha decay".  
10.2 12/2 Derivation of path integral representation of transformation function.
Some brief comments on propagators.
Notes .
Read Sakurai p109-116 on propagators. I will not cover all this in lectures but expect you to have read it.
For further reading on path integrals see not only Sakurai but also Gottfried & Yan and the classic, Feynman & Hibbs ("Quantum Mechanics and Path Integrals", McGraw-Hill, 1965). Also good is the first chapter of Lowell S. Brown, "Quantum Field Theory" (Cambridge, 1992).  
10.3 12/3 Calculating the SHO transformation function using path integrals.
Notes .
Feynman and Hibbs.  
11.1 12/7 E and B fields in classical mechanics.
E and B fields in QM---Heisenberg equations of motion, Landau levels, probability current.
Begin gauge transformations in QM.
Notes .
Read Sakurai, section 2.6, which gives a nice discussion. We will not cover all the material in this section, but you will delve into Landau levels more deeply in HW8.  
11.2 12/9 More on gauge transformations in QM.
Aharanov-Bohm effect (Refs for experiments: R.G. Chambers, Phys. Rev. Lett. 5 (1960) 3 and A. Tonomura et al, Phys. Rev. Lett. 48 (1982) 1443.)
Notes .
Read in Sakurai 2.6 about gravitationally induced quantum interference
(Experimental refs: R. Colella et al, Phys. Rev. Lett. 34 (1975) 1472 and H. Kaiser et al., Physica B385 (2006) 1384.)  
11.3 12/10 Review for final exam  
12 12/15 FINAL EXAM (10:30-12:20) Solution is here .

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Stephen Sharpe
Last modified: Fri May 23 11:56:53 PDT 2008