PHYS6010 3.0 Website for F2008

Class meets Thursdays 12:30-2:30, location: Farquharson 045; small time adjustment will be applied to help students going to/from Bethune before/after class for PHYS5020.

On Thursday, February 19 (last class), I will discuss homonuclear diatomic systems, so that you can read articles on determining the potential between atoms (which are of interest in cold atom trap experiments, and ultimately in BEC). If you wish to prepare for the class, looking in the Quantum Chemistry book by Ira Levine, Chapter 13 would help. One such method of measuring the details is called photoassciation spectroscopy, a review article is linked here:

Reviews of Modern Physics 78, p.483

This paper I have included mostly as a recent reference on the experimental efforts to pin down the details of atom-atom interactions relevant for BEC. In class I will go over some parts of the paper listed below. You should look at it before class. It connects with some of the ideas on the behaviour of phase shifts at low energy which we dealt with in previous classes.

A theoretical paper on modeling the atom-atom potential for alkali dimers is here; it also shows how to calculate the scattering length:

Physical Review A 48, p.546

Review Paper by Dalibard on cold atom scattering: Dalibard; Theoretical paper on Li-Li scattering: Phys Rev A 50, p. 399

Notes from the last class: atom-atom potential curves

Reading Assignment for Thursday, February 12> We will discuss scattering from a Lennard-Jones potential. The worksheet (in New Maple format, *.mw) is here: For those who prefer the classic style (mws) it is here, but it misses some graphs DifferentialCrossSectionLJ.mws. The reading is from M.S. Child, pp. 60-77, which, however, puts emphasis on recovering quantum mechanics from the classical results (which we won't go into). Concentrate on the results. Your next assignment will be to write your own summary of what scattering from a LJ potential is all about.

Total cross section (TCS) for a LJ potential Maple-mw worksheet:

For the final assignment (mostly literature work, but on the basis of understanding scattering from a Lennard-Jones potential, i.e., DCS and TCS) I will post some research articles with experimental results here.

Proton-Rare Gas Scattering, Phys Rev A 2, 2327 (1970) p-Ar,Ne

For students more interested in atom-atom scattering: you can write up something on determining the scattering length for cold atom-atom collisions based upon the last class, and upon the articles posted above. The value of the final assignment (5-10 pages of text including graphs) will be 30% towards the course.

Now (end of Jan) that classes are resuming: Here is a maple worksheet that solves the classical scattering problem (b-theta relationship and differential cross section), and illustrates it for the exponential potential. Play with the worksheet to understand classical scattering. The next worksheet which I will post will look at the quantum scattering cross section based upon WKB phase shifts. In addition, there is a more general worksheet that illustrates classical scattering from a Lennard-Jones type potential (rainbow and glory)



The effects from atomic scattering as contained in a phenomenological Lennard-Jones potential are discussed in M.S. Child's book on pp.65 ff.

Reading Assignment 5 for November 6: up to p.63, but scan only between eqs (4.24-4.37). Concentrate on the Classical Limit section, pp.59-63. From the previous reading (Oct 30), there is a short derivation to be handed in (see below)

For the class on October 23, please look at the new version of PhaseShiftNum.mws, which is given as html here (a browser displays the output): PhaseShiftNumerical

The worksheet for the numerical calculation of phase shifts for arbitrary potentials which I demonstrated in the Sept 25 class is here: PhaseShiftNum.mws It is considerably expanded to look at the connection bewtween bound states supported by the potential and the scattering length. Will be discussed on Oct 23.

Second hand-in assignment (extended to Oct 2) consists of a graphical demonstration of how the partial wave expansion works for the plane wave, eq. (1.23) on page 30. I am thinking of the following: pick simple units (like discussed in class), choose k=0.5 and k=1 (to have at least 2 examples), calculate the real and imaginary parts on both sides of the equation (express z in terms of r and theta), so for fixed k this will depend on two independent variables. The right hand side can be computed for truncated sums, cut off the sums, e.g., after 10 terms, 50 terms, whatever you feel is appropriate. Normal 2d plots may suffice, e.g., you pick a theta value, and graph as a function of r. Make your observations when comparing the exact left-hand side to the approximate right-hand side, and write them up.

Third hand-in assignment (Oct 16) Use the Maple worksheet from Computational Physics using Maple #5.13 to investigate the convergence properties of the total cross section as a function of wavenumber (or incident particle energy k^2/2 when hbar=1) for an attractive or repulsive square-well potential of your choice [different from the one used in the worksheet]. For two wavenumber values (energy values) - one lower, one higher look also at the convergence properties for the differential cross section (you may want to start with that part).

Fourth hand-in assignment (Nov 6) Derive Eq. (4.7) on p. 53, and verify that W is the Wronskian. Use the internet (or Arfken) to find out the meaning of the Wronskian in differential equations.

Fifth hand-in assignment (after Nov 6): Look at the problem of computing eigenenergies and eigenfunctions of an attractive exponential potential. The exact s-wave eigenvalues can be obtained from solving a nonlinear equation (details are problem 10.8 in Quantum Mechanics by Ghatak and Lokanathan, QC174.12 G485 2004 - they explain that a transformation turns the s-wave Schrodinger eqn with the exp potential into a Bessel equation, and the demand that the solution be regular at the origin yields the bound-state eigenvalue condition):

BesselJ(nu,g)=0 is to be solved where in simplified units (hbar=1, mass=1, length scale a=1) g = sqrt(8*mu*V0) measures the strength of the exponential potential -V0*exp(-r/mu), and this is V, not U. Graph the Bessel function and realize that it has no root as a function of the 'real-valued' order nu when g<2.405, then one root, and find the critical g-value when two roots appear (2nd s-eigenenergy) as g is increased (i.e., V0 is bigger, a deeper potential). The eigenenergies are obtained from nu using E = -nu^2/(8*mu). Just work with mu=1. For example, for our matrix diagonalization calculation presented in the worksheet (last class, see website), we had V0=1, g=2.828, the root nu is found to be 0.2824, yielding E0=-0.009970. This is below what was found in the matrix diagonalization (-0.00970). The exact eigenfunction is given as N*BesselJ(nu,g*xi), where N is a normalization constant, and xi=exp(-r/2).

Assignment question: Pick a value of the strength parameter (start with V0=1, but then go and choose your own individual value, preferably such that two negative-E eigenvalues appear). Compute the exact eigenvalues by finding the roots of the Bessel-equation above. Then play with the matrix eigenvalue code by choosing an appropriate box size, and by varying the basis size (N=10, 20 , 30) and observe the convergence of the two bound-state eigenvalues against the exact result. Investigate the pseudo-convergence, i.e., convergence against a wrong energy when the box size is too small. Comment and explain your results! To observe the effect of box size look at the exact eigenfunctions.

Sixth hand-in assignment (Dec 3): download the worksheet PhaseShiftCalculator2.mws. Work through it and carry out the assignment at the end. The worksheet will give you practical exposure to a few results from your readings such as the WKB result for the phaseshift (Eq. (4.23) on p.56), its high-energy approximation (Eq. (4.38) on p.59). The next step will be to really understand the 'naive' semi-classical result Eq.(4.47) on p. 62., and how to calculate differential cross sections by integrating over L in the complex plane. I am working on a worksheet for that, and it will lead to your seventh assignment. This will make up for a few weeks of missed classes. A better discussion of this topic is given in M. Child, chapter 5.


This will keep you busy during the lecture-free time.

The wavepacket demonstration from the Sept 18 class was from then go for section 5.5 If numerical wavepacket propagation interests you, and how such a wavepacket when moving into a potential region comes to a halt and turns around, look at 5.8.

This is a course on the quantum theory of scattering with a heavy focus on atom-atom scattering in the final third of the course.

Prerequisites: PHYS5010, or PHYS5050, or equivalent, or permission from the course director.

Marking scheme:

about 8 assignments, some with computational aspects (done in maple, or mma, or matlab): 30% (they will happen up until the 2/3 point of the course)

Midterm exam: 30% (will cover the general scattering theory parts, and will happen at the 2/3 point of the course)

Presentation: 20% (based on a research publication on low-energy atom-atom scattering, topic to be confirmed with instructor); students participate in the evaluation process.

Final quiz on the topics covered in student presentations: 20%

Literature: We start with the text by L.S. Rodberg and R.M.Thaler which is provided in electronic format (deja vu), later we will use the book by M.S. Child on Molecular Scattering (and the semiclassical approximation). That book, and the text by C.J. Joachain on Quantum Collision Theory will also be provided as djvu files on this website. Students will need to provide themselves with a djvu reader, such as WinDjView (freeware). The library hardcopies will be put on 1-day reserve. As far as I know the books are out of print - that's why putting the djvu files here isn't such a bad thing.

Rodberg-Thaler (djvu).

Child (djvu).

Joachain (djvu).

A website where WinDjView or a Mac version (MacDjView) can be found: