Individual (or team) Projects

Computational Quantum Mechanics, Physics 498CQM, Spring 2001

The projects are an important part of this class and will be the key way that each student becomes involved in a topic at a depth beyond what is possible in the lectures and assignments. It can be a topic at a more advanced research level, a study of a conceptual issue in quantum mechanics, an issue in computation related to quantum mechanics, or constructing software for illustrating principles and teaching quantum mechanics.

I would like a one page 'proposal' from each student by the end of February. Please get in touch with me concerning your plans for a project before that date. You are encouraged to pair up with another student in the class on the project; of course, in this case the level of the project should be higher than for an individual project.

Each project should be described in class in a roughly fifteen-minute presentation in the latter part of the semester. I will schedule talks with individuals, so that they will be spread over the last month.

Each project should also be described in a write-up in Revtex which is brief, but sufficient for describing the key ideas. (This form is important to learn since it is the form in which you would write a paper for Physical Review; It is not difficult to become familiar with the rudiments of Latex and Revtex so that you can use them.) Figures should be in electronic form as well. Some of the projects listed may be publishable if you do extra work. I will put the written projects with figures on the class Web site.

Here I give some suggestions for projects. I can supply references to help you get started on these projects. This list is NOT meant to be exhaustive. I will be very pleased for you to suggest alternative projects! The projects may involve a significant modification or optimization of an existing code or algorithm; a new implementation, such an interesting issue in science; the use of new computer architectures; a conceptual issue in quantum mechanics, such as quantum mechanics of simple systems coupled to a dissipative bath; research in the literature and writing an algorithm related to the new area of "Quantum Computing"; or development of software for teaching quantum mechanics.

1. Starting from codes in Koonin's projects or our class projects, you could make a significant modification. Possible examples:
   • Modifying the QMC algorithm to treat Fermions with the "sign problem"
   • Improve the plane wave band structure code to do self-consistent density functional calculations on crystals.
   • Modify the plane wave band structure code treat the "photonic bands", i.e., states of electromagnetic waves in inhomogeneous media. This is an active research area now.
   • Modify the shell model code to treat more realistic model for nuclear interactions and carry out example calculations.
   • A project using the Lanczos algorithm for a many-body problem

2. Use (or modify) a program which I have (density functional calculation for crystals, group theory analysis, lattice Monte Carlo) for a calculation. In this case it will be your responsibility to learn the use of the program.

3. Construct a program for evolution of the time-dependent Schrodinger equation that could be used in teaching quantum mechanics.

4. Construct a "Car-Parrinello" program to simulate quantum systems with molecular dynamics (or related) methods. This would be to go beyond what is possible in class.

5. Study Bose-Einstein condensation. How can this be identified by computations? Read about and apply the ideas of "stiffness" to changes in boundary conditions.

6. In variational Monte Carlo, try doing a random walk in parameter space to find the minimum energy or variance. Try to determine good rules for moving the parameters. Alternatively, is the energy minimization or the variance minimization sharper? Which has the smallest errors for the optimal parameters?

7. Elementary particles and lattice gauge theory. Analyze the recent advance reported by Lepage and coworkers of Cornell, who say that by rearranging perturbation theory in the lattice versions of the continuous theory, they are able to use larger lattice spacing and speed up the calculations by many orders of magnitude. (See report in Science 270, 1757 (1995).) Because I am not as familiar with this area, I would like to learn!

8. Carry out a Lanczos or Monte Carlo calculation for a problem in elementary particle theory.

9. Solving Eliashberg equations for superconducting transition temperature in a model superconductoring a method that would illustrate computational methods applied to self-consistent equations.

10. Calculation of moments of densities of states using recursion and/or maximum entropy methods. These can be very useful in calculating local densities of states and Greens functions.

11. Review recent work on perturbation theory and the "2n + 1" theorem, and carry out calculations in a model example.

12. Implement an order(N) method (it goes like N asymptotically) for computing electronic states. This is an active research area in which my research group is involved.

13. Review the literature on "functionals" and give examples of uses beyond that done in class.

14. Review recent papers on "Quantum Computing", possibly on work showing that factorization can in principle be done by a quantum computer that could not be done on a classical one. Construct an algorithm to simulate a "Quantum Computer". See for example the review "Quantum Computing and Shor's Factoring Algorithm", A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996).

15. Review ideas of Berry's phases in quantum systems and work out an example.

16. Review the use of Berry's phases to calculate electric polarization in crystals, and carry out simple calculation. See for example the review by Resta in Rev. Mod. Phys. 66, 899 (1994), which has fererences to the original Berry article and other papers.

17. Construct a general tight-binding program that would calculate bands and total energy for general crystals.

18. Construct a group theory program that would automatically determine group operations for a molecule and crystal. If an existing program were used, it could modified or applied in a novel way.

19. Research the literature and write a program to do a many-body "GW" or "FLEX" quasiparticle calculation. There exist remarably simple approximations which work well in solids and could be used in a calculation.

20. A project using the ideas of large degeneracy and large dimension to simplify many-body calculations, and construct an algorithm for a simple case.

Return to 498CQM Home Page