Gisbert Wuestholz, Eidgenössische Technische Hochschule (Zurigo, Svizzera)
Modern transcendental methods in Diophantine Geometry
Sunday, November 21 2004, at 17:00
Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza | |
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Dennis Weaire, Trinity College (Dublino, Irlanda)
The many facets of foam
Friday, October 01 2004, at 15:00
Dipartimento di Matematica - Politecnico di Milano - Via Bonardi 9 - Milano - Aula Consiglio (VII piano) | |
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José C. Geromel, UNICAMP (Campinas, Brasile)
H2 optimal robust filtering
Wednesday, July 07 2004, at 17:00
Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza | |
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Alexander N. Sharkovsky, National Academy of Sciences (Kiev, Ucraina)
Ideal turbulence: definition and models
Monday, July 05 2004, at 17:00
Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza | |
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Shige Peng, Shandong University, Jinan (Cina)
Continuous Time Risk Measures and Evaluations by BSDE
Monday, June 14 2004, at 17:00
Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza | |
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Wu-Yi Hsiang, Hong Kong University of Science and Technology (Hong Kong, Cina)
Three body problems in quantum mechanics
Wednesday, May 26 2004, at 17:00
Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza |
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Abstract
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In this talk, I shall describe the geometric approach to solve the Schrödinger equation for various physically meaningful three body systems such as He, H2+, H-, three bosons in R2 with d-function potential etc. The configuration space of the three body system in R3(resp. R2) (with center of gravity fixed at the origin) is an R6 (resp. R4) equipped with an SO(3) (resp. SO(2)) symmetric kinematic metric, while the potential function U is also SO(3) (resp. SO(2)) invariant. The first step is to fully utilize the SO(3) (resp. SO(2)) symmetry to reduce the Schrödinger equation to an equation solely defined at the level of the orbit space (i.e. R6/SO(3) (resp. R4/SO(2))) equipped with the orbital distance metric. One needs to make effective use of both group representation theory and equivariant differential geometry to achieve such a reduction. The orbit space of a three body system in R3 (resp. R2) equipped with the orbital distance metric is always isometric to the Riemannian cone over S2+ (1/2) (resp. S2(1/2))), namely the Euclidean hemisphere (resp. sphere) of radius 1/2. This remarkable fact (i.e. sphericality) enables us to bring in the spherical harmonics and their generalizations (namely, Jacobi polynomials and monopole harmonics) to greatly simplify the analysis of the angular part of the reduced equation. I will use the simpler case of the boson system to illustrate this step which enables us to further reduce the Schrödinger equation to an ODE solely in the radial direction. Such an ODE can be thoroughly analyzed and I will discuss the physical significance of these solutions so obtained for the three boson system. Bibliography Wu-Yi Hsiang. Kinematic geometry of mass-triangles and reduction of Schr¨odinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters. Proc. Nat. Acad. Sci. U.S.A., 94(17):8936–8938, 1997. Wu-Yi Hsiang. On the kinematic geometry of many body systems. Chinese Ann. Math. Ser. B, 20(1):11–28, 1999. A Chinese summary appears in Chinese Ann. Math. Ser. A 20 (1999), no. 1, 141. |
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