薛丁格方程在雙子束縛態的推廣：庫伯對與激子上的應用

Generalization of Schrodinger equation to two-body bound state : its application on Cooper pair and exciton

蕭銘翰(Ming-Han Hsiao)

指導教授 ： 陳正弦

共同指導教授 ： 林倫年

國立臺灣大學/理學院/物理研究所/博士(2011年)

https://doi.org/10.6342/NTU.2011.00291

摘要

對於不同材質的光譜分析研究一直是物理領域中很重要的課題，主要難處在於是否可以完善處理所給定的固態或分子材料中的複雜交互作用。不論在理論上或計算上，這都相當地具有挑戰性，多年來一直吸引著許多偉大的物理學家進入並提出各種的貢獻，發展出許多實用的工具與技巧。
本論文中我提出了對於薛丁格方程式在雙子束縛態的推廣，以及相對應的格林函數。首先我將其應用在超導領域中與凝態領域中的雙子束縛態，即庫伯對以及激子，其中對激子的研究其實是觸發我這篇論文的動機。論文中將說明此方程式的推導與物理意義，上述兩例應用的結果與預測也同樣呈現在論文中。
關於此理論還需要更多的探索。本工作是個起點，未來我的方向是繼續此理論推廣至二次量子化，即場論的領域，要強調的一點是此二次量子化必須相容於格林函數的運動方程。我希望達到的目標是讓場論可以有不同於平面波展開的形式，物理意義類似於有一派理論提說的薛丁格場。依此，場論可以有更豐富的內涵與應用。

關鍵字

雙子束縛態 ； 激子 ； 強作用超導體

並列摘要

The investigation of optical spectrums among various materials has been a major task for physical research, mainly to deal with the complex interaction among electrons,
the so-called exchange-correlation kernel, within some given solids or molecules. Both theoretically and numerically this is by all means a very challenging problem, which attracts many great minds into the regime, and many useful techniques and powerful tools have been developed for decades.
In this work I propose a two-bound extension version of Schrodinger equation, as well as the Green function correspondingly. As a first attempt, this work aims to apply the method to some well-known two-body bound state, the Cooper pair in BCS theory and the exciton in BSE, where BSE in solid state society is in fact the very equation which triggers me in the first place to get into this way of thinking. The derivation of the method proposed is given, and the results and predictions are shown in the text.
The method may require further study, and actually this is just a starting point. In the future, I wish to develop the method up to the quantum field regime, which means the
second quantization scheme is needed which should be consistent with the equation of motion of Green function and in this sense, the quantum field theory could gain more
physical interpretation and be generalized to some extent more than its usual plane-wave form. This is actually in line with the thought of so called Schrodinger field.

並列關鍵字

two-body bound state ； exciton ； strong coupling superconductor

參考文獻

chapter 1 references :

[1] L. Hedin, Phys. Rev. 139, A796 (1965).

[2] L. Hedin and S. Lundqvist, in Solid State Physics, Advances in Research and Application, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, New York,

[3] P. Hohenburg, W. Kohn, Phys, Rev. 136, B864 (1964).

[6] W. Kohn and L. J. Sham, Phys, Rev. 140, A1133 (1965).