Quais-Exact Solvability of Charged Particles in External Fields

• Main Authors: Chun-Ming Chiang, 江俊明
• Other Authors: Choon-Lin Ho
• Format: Others
• Language: en_US
• Published: 2002
• Online Access: http://ndltd.ncl.edu.tw/handle/55447465698561314633

博士 === 淡江大學 === 物理學系 === 90 === Title of Thesis: Total pages: 102 Quasi-Exact-Solvability of Charged Particles in External Fields Key Words: quasi-exactly solvable (QES) model; Bethe ansatz approach Name of Institute: Department of Physics, Tamkang University Graduate Date: Jan. 2002 Degree Conferred: Doctor of Philosophy Name of Student: Chun-Ming Chiang Advisor: Professor Choon-Lin Ho 江 俊 明 何 俊 麟 Abstract: We study a new type of spectral problem, the so-called quasi-exactly solvable problem. This is a special class of quantum-mechanical problems for which analytical solutions are possible only for parts of the energy spectra and for particular values of the fundamental parameters. Four physical examples of quasi-exactly solvable models are presented, namely, (A) two-dimensional electron in Coulomb and magnetic fields, (B) relativistic two-dimensional electron in Coulomb and magnetic fields (the Klein-Gordon case), (C) relative motion of two electrons in an external oscillator potential, and (D) the motion of a Dirac electron in two spatial dimensions in the Coulomb and magnetic fields. We show exact recursion relations that determine the coefficients of the series expansion of wave functions, the possible energies, and magnetic fields. It is found that analytic solutions are possible for a denumerably infinite set of magnetic field strengths. In this work we display a purely algebraic solution based on the Bethe ansatz equations. We also demonstrate that the Bethe ansatz approach yields the same spectrum as that obtained by solving recursion relations. We show that these cases are reducible to the same basic equation, with the exception of the Dirac case. These cases (A, B, and C) can be given a unified treatment, which is quasi-exactly solvable owing to the existence of a hidden sl2 algebraic structure. Finally, we show that the quasi-exactly-solvability of the system (D) is not related to the sl2 algebraic structure.