Application of a transformed expansion method to perturbation theory and supersymmetric quantum mechanics

• Main Authors: Chia-Chun Chou, 周佳駿
• Other Authors: Ching-Teh Li
• Format: Others
• Language: en_US
• Published: 2003
• Online Access: http://ndltd.ncl.edu.tw/handle/37595385141710544540

碩士 === 國立臺灣大學 === 物理學研究所 === 91 === Perturbation theory is based on the assumption that the problem we wish to solve is, in some sense, only slightly different from a problem that can be solved exactly. The desired quantities are expressed as power series expansions in powers of a perturbation parameter. The first few terms of the power series expansion do properly describe physical systems. Unfortunately, such a power series expansion is usually strongly divergent even if the value of the perturbation parameter is small. In this thesis, we attempt to solve the energy eigenvalues of the one-dimensional anharmonic oscillators approximately by a transformed expansion method. We transform the power series that is proper when the perturbation is small into a new power series that preserves the correct functional form when the perturbation is large. We introduce an unphysical parameter into the new power series expansion in the process of the transformation. The value of the unphysical parameter is determined by the principle of minimal sensitivity. By applying the method to perturbation theory, we can obtain fairly accurate results for energy levels even if the anharmonicity is large. Furthermore, we also apply the method to supersymmetric quantum mechanics. By making good use of the properties of supersymmetric quantum mechanics, not only can we obtain fairly accurate results for energy levels but we can also get approximate wavefunctions. The method proposed here may be useful for solving similar quantum mechanical problems.