Bulletin of the American Physical Society
2008 APS April Meeting and HEDP/HEDLA Meeting
Volume 53, Number 5
Friday–Tuesday, April 11–15, 2008; St. Louis, Missouri
Session J4: Topics in Mathematical and Computational Physics |
Hide Abstracts |
Sponsoring Units: DCOMP Chair: Rubin Landau, Oregon State University Room: Hyatt Regency St. Louis Riverfront (formerly Adam's Mark Hotel), Promenade B |
Sunday, April 13, 2008 10:45AM - 11:21AM |
J4.00001: Edward A. Bouchet Award Talk: NSFD Schemes: Genesis, Methodology and Applications Invited Speaker: Nonstandard finite difference (NSFD) schemes are based on a generalization of the usual discrete representations of first derivatives and the use of nonlocal discrete replacements for both linear and nonlinear functions of dependent variables. These numerical integration techniques for differential equations had their genesis in a 1989 publication.$^{1)}$ In the past decade much progress has occurred on the general methodology of these techniques and the range of phenomena to which these schemes have been applied.$^{2)}$ This talk will give a broad introduction to NSFD schemes and show that the principle of dynamic consistency (DC)$^{3)}$ can be used to place great restrictions on the constructions of such discretizations for both ODE's and PDE's. The essential features of the NSFD methodology will be illustrated by means of several ``toy" models.$^{4)}$ \newline \newline $^{1)}$R. E. Mickens, {\it Numerical Methods for PDE's, \bf 5} (1989), 313--325. \newline $^{2)}$K. C. Patidar, {\it Journal of Difference Equations and Applications \bf 11} (2005), 735--758. \newline $^{3)}$R. E. Mickens, {\it Journal of Difference Equations and Applications \bf 11} (2005), 645--653. \newline $^{4)}$R. E. Mickens (editor), {\it Advances in the Applications of Nonstandard Finite Difference Schemes}. World Scientific, Singapore, 2006. [Preview Abstract] |
Sunday, April 13, 2008 11:21AM - 11:57AM |
J4.00002: Ghostbusting: Reviving quantum theories that were thought to be dead Invited Speaker: The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (symmetric under combined matrix transposition and complex conjugation) in order to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, for example, which is clearly not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a perfectly acceptable quantum mechanics. Evidently, the axiom of Dirac Hermiticity is too restrictive. While the Hamiltonian $H=p^2+ix^3$ is not Dirac Hermitian, it is PT symmetric; that is, it is symmetric under combined space reflection P and time reversal T. In general, if a Hamiltonian $H$ is not Dirac Hermitian but exhibits an unbroken PT symmetry, there is a procedure for determining the adjoint operation under which $H$ is Hermitian. (It is wrong to assume a priori that the adjoint operation that interchanges bra vectors and ket vectors in the Hilbert space of states is the Dirac adjoint. This would be like assuming a priori what the metric $g^{\mu\nu}$ in curved space is before solving Einstein's equations.) In the past a number of interesting quantum theories, such as the Lee model and the Pais-Uhlenbeck model, were abandoned because they were thought to have an incurable disease. The symptom of the disease was the appearance of ghost states (states of negative norm). The cause of the disease is that the Hamiltonians for these models were inappropriately treated as if they were Dirac Hermitian. The disease can be cured because the Hamiltonians for these models are PT symmetric, and one can calculate exactly and in closed form the appropriate adjoint operation under which each Hamiltonian is Hermitian. When this is done, one can see immediately that there are no ghost states and that these models are fully acceptable quantum theories. [Preview Abstract] |
Sunday, April 13, 2008 11:57AM - 12:33PM |
J4.00003: Nicholas Metropolis Award for Outstanding Doctoral Thesis Work in Computational Physics Talk: Equation of State of the Dilute Fermi Gases Invited Speaker: In the recent years, dilute Fermi gases have played the center stage role in the many-body physics. The gas of neutral alkali atoms such as Lithium-6 and Potassium-40 can be trapped at temperatures below the Fermi degeneracy. The most relevant feature of these gases is that the interaction is tunable and strongly interacting superfluid can be artificially created. I will discuss the recent progress in understanding the ground state properties of the dilute Fermi gases at different interaction regimes. First, I will present the case of the spin symmetric systems where the Fermi gas can smoothly crossover from the BCS regime to the BEC regime. Then, I will discuss the case of the spin polarized systems, where different quantum phases can occur as a function of the polarization. In the laboratory, the trapped Fermi gas shows spatial dependence of the different quantum phases. This can be understood in the context of the local variation of the chemical potential. I will present the most accurate quantum ab initio results and the relevant experiments. [Preview Abstract] |
Follow Us |
Engage
Become an APS Member |
My APS
Renew Membership |
Information for |
About APSThe American Physical Society (APS) is a non-profit membership organization working to advance the knowledge of physics. |
© 2024 American Physical Society
| All rights reserved | Terms of Use
| Contact Us
Headquarters
1 Physics Ellipse, College Park, MD 20740-3844
(301) 209-3200
Editorial Office
100 Motor Pkwy, Suite 110, Hauppauge, NY 11788
(631) 591-4000
Office of Public Affairs
529 14th St NW, Suite 1050, Washington, D.C. 20045-2001
(202) 662-8700