** PHYS 5000 3.0 AF Quantum Mechanics**

** PLEASE NOTE:** Grades are due by January 12, 2010. If you have outstanding work
for me, please hand it in on Monday, Jan. 11, before 3pm, so that it can count.

Classes: Tuesdays 11:30-1pm in PSE321; Thursdays 10-11:30 in PSE 317 (11-11:30 question time). Start: Thursday, Sept 17.

Maple worksheet from the Sept 17 class: PH5000L1.mw ; for older maple versions or for unix users: PH5000L1.mws . You can open these only with maple software!

** Assignment 1:** Expand the worksheet to add a solution which is antisymmetric [ICa:=psi(0)=0,D(psi)(0)=1;]. Observe the closeness of the level spacings for the choice of potential [look at (E1-E0)/(E1+E0)]. Modify the potential parameters such that E0 and E1 agree to 2-3 digits (near-degeneracy). Generate graphs of the symmetric (E0) and anti-symmetric (E1) solutions for such a case, label them with the potential parameters of your choice, and compare the wavefunctions. Construct the probability densities [this will require properly normalized wavefunctions: you can calculate the norm integrals approximately as h*add(psi(n*h)^2,n=-N..N) , where N is sufficiently large - not too large, such that you are restricting the x-values to be within the range where psi(x) touches the axis at large r; choose h sufficiently small for reasonable accuracy]. Do not submit your maple worksheet. Submit a write-up with the graphs, and some comments about the similarity of the probability density for both states. You can also look at the states generated by adding/subtracting the properly normalized psi0 and psi1. Addition/subtraction is justified by the near-degeneracy of the eigenenergies. Comment on the meaning of these linearly combined states.

Due date: October 1st, in class (if possible).

Maple worksheet from the Sept 24 class: PH5000L2.mw for unix/classic users: PH5000L2.mws .

Completeness Relation example: Completeness.mw ; Completeness.pdf

Maple worksheet from the Oct 1 class: PH5000L3.mw for unix/classic users: PH5000L3.mws .

** Assignment 2:**
Use the worksheet from the Oct 1 class (link is above), and set the HO basis up such that it works well for one of the near-degenerate pairs of eigenstates (pick n=0/1, or 2/3, or 4/5 for the double-well potential V). This means: pick N=40, and make adjustments to the
HO spring constant (w or beta=sqrt(w)). Now set up the time evolution for one of the two one-sided states PsiL or PsiR (using the corresponding eigenvalues for the near-degenerate pair). Plot the probability density for a sequence of times such that you observe the motion of probability density from one side to the other and back. Hint: the time scale over which the flow happens has something to do with 1/(E_{n1}-E_{n2}), where E_{n1} [E_{n2}] is the energy of the higher [lower] of the near-degenerate pair of states.

Maple worksheet from the Oct 8 class: PH5000L4.mw for unix/classic users: PH5000L4.mws.

**Assignment3:** Use the above worksheet to investigate tunneling through a double barrier. Some notes as to how the method works (explained in class) can be found in my Computational Physics using Maple webpages.
Pick a double barrier of your choice. Calculate transmission coefficients for some set of energies (wavenumbers) below and above the barrier height. Also compute them for the corresponding single barrier. Observe whether there are energies for which the transmission probability for a double barrier can be understood as the product of probabilities to go through a single barrier.
Compute the eigenvalue for the bound state which is formed when the barrier height is extended on both sides to very large x (positive and negative). For this you can use the very first worksheet (assignment 1).
Compare the transmission resonance position to this energy. How far is the resonance peak (T=1) away from the eigenvalue position? How wide is the resonance structure in comparison?
If you have time and patience: repeat the calculation for a similar double barrier, but with thicker barrier widths. The resonance structure should be narrower. Is the energy difference between transmission peak and bound-state position also closer?

Reading material for the week of Oct 19: From the book by John S. Townsend, A Modern Approach to Quantum Mechanics (QC 174.12 T69 2000), the section on partial wave analysis. PWA1.PDF

Review article for cold atom-atom collisions: Julienne.pdf

Maple worksheet from the Oct 22 class: PH5000L5.mw

Maple worksheet from the Oct 29 and Nov 5 classes: PH5000L6.mw;

** Assignment 4:** Investigate Levinson's theorem for a partial wave of L>0 in the attractive exponential potential. Choose your value of V0 (strength), and use the worksheet PH5000L5.mw to calculate the phase shift eta_L(k) for your choice of L=1 (p-wave) or L=2 (d-wave), and find out the correct value of eta_L(k=0) (an integer multiple of Pi) such that eta_L(k->infinity) goes to zero. Plot the partial-wave cross section and indicate where it reaches the unitarity limit. Then use PH5000L6.mw to investigate the bound-state energies. First, understand for L=0 (for which an exact solution can be obtained from Bessel's equation) how the matrix representation converges (number of significant digits for a given matrix size, you don't need to reach very high accuracy, just state how accurate your results are, ie. don't go overboard on matrix size and computing time!). Then apply your experience to your chosen L-wave (for which you'll have no exact benchmark eigenvalues), and determine the bound states (again, high accuracy is not that important, but you need to state how confident you are about the result by comparing results from neighboring matrix sizes, and by optimizing the beta-parameter in the Legendre-basis). Does the number of bound states match up with your findings from the scattering calculation?

Maple worksheet from the Nov 12 class: PH5000L7.mw

Suggested readings for the Tuesday Nov 19 and following classes will be from the textbook by W. Greiner (a djvu reader is required). GreinerRQM.djvu

For Nov 17, remind yourselves about relativistic notation, Sections 1.1-1.4.

For Nov 19 pre-read sections 1.6-1.8.

For Nov 24 pre-read sections 1.9-1.10.

Article by R. Winter on the Klein Paradox referenced in the hand-out: Winter.pdf (Am. J. Phys. 27, 355 (1959).

Remark about Nov 24 sign ambiguity: The correct factor that goes with phi(r)^2 is 2*(E-e*V(r)) where V(r) is the potential, and e*V(r) the potential energy. See (4.23) in the hand=out.

On Dec. 1 (Tuesday class) we will have our midterm based on the material that went into the four computational assignments. Bring your lecture notes and assignments. Also look up how to calculate the phase shifts for a spherical potential well/barrier (most QM books will have it).

**Final problem sets (due in early January).**

1) Join a group to do either Exercise 1.14 (p. 56 in Greiner), or Exercise 1.15 (p. 59). Email me the choice of problem, and I will announce who is working together. I expect a few-page hand-in, with your own plots demonstrating the Figures in Greiner, and some other parameter choice.

2) also group work: work through Example 9.3 (p. 210 in Greiner) to derive the radial Dirac equation(s) for a central potential. Then solve them (Eqs 26) numerically by a shooting method (Maple or some other software) for a hydrogenlike problem with charge Z for the[group A: ground state (1S), group B: lowest P state]. Demonstrate how the role of the small component increases with Z. Hand-in: report on the numerical work.

**Course Syllabus**

This course is a continuation of your undergraduate quantum mechancis education. Students are expected to have completed 3-4 semesters of modern physics/quantum mechancis courses at the undergraduate level, and will have covered most topics as presented in a textbook such as Richard Liboff's.

We begin with a brief review of the matrix representation of quantum mechanics, and computational methods to solve eigenvalue problems using both the differential-equation and the matrix eigenvalue problem. We will also touch briefly upon the scattering problem (phase shift analysis). This will serve as a reminder, and as an introduction to computational methods (Maple will be supported, other computing tools can be used, there will be computational assignments to deepen the understanding.

Then we will look at (cold) atom-atom interactions via hyperfine forces. This will be based upon recent research articles. Alternatively, the class decides on a choice for this quarter of the course based on some topic suggestions given below

A large part of the course will be devoted to relativistic quantum mechanics. For this part we will use the textbook of W. Greiner (RQM: Wave Equations), which will be made available in electronic form.

If you are unfamiliar with any computational tools, you can learn Maple on your own, and you can make use of an extensive collection of worksheets from my Computational Physics using Maple site linked under www.yorku.ca/marko I can provide some support with Matlab and Fortran, for other environments you will be on your own. Maple can be purchased for $100 from CNS.http://www.yorku.ca/computing/facultystaff/software/grouplicense/index.php They are currently still selling Maple12, but should move to Maple13 very soon.

**Marking Scheme**

10 Assignments worth 5 % each, six of the assignments computational, four will be pencil/paper type

Midterm examination: 20 %

Final examination: 30 %