應用於量子系統中的微正則系綜數值方法與廣義邊界微擾變分方法

Abstract

在本論文中主要包含三部份的工作: (1) 第一部份是關於一套我們所 發展出來數值方法。在計算物理的領域裡﹐費米子的反互換( anti- commutation ) 特性一直是長久以來無法被妥善處理的問題。關於這個難 題﹐我們以統計力學中的微正則系綜理論為基礎﹐提出並證明一套可以克 服它的新理論。在我們的方法中﹐特點是使用量子系統的能量期望值而非 能量本徵值做為系統的能量定義。 (2) 在論文的第二部份﹐我們改進 Gorecki 與 Byers Brown的方法成為廣義的變分方法。在他們的方法中變 分函數定義為 ﹐其中 是被用來變分的輪廓( contour )函數。他們的方 法只能計算受侷限系統的基態。而且﹐由於 不易做形狀變分﹐所以他們 的方法對於不對稱系統的計算結果不是很令人滿意,也造成他們的方法不 常被使用。我們利用基礎函數的線性組合方式來廣義化他們的方法。我們 所用的變分函數為 。其中 就是基礎函數的線性組合。如此﹐我們的方法 可以有效率地模擬不對稱系統。而且﹐我們的方法可以用來計算激發態。 (3)論文的第三部份是我們將研究方向轉到計算物理之前的早期工作。我 們系統地研究一個簡化的BCS模型。我們推導出此模型的能量本徵函數。 這些能量本徵函數具有很明顯的對稱性破壞性質。其中基態與激發態分別 具有不同的玻色子與費米子行為。這些結果有助於對稱性破壞現象的瞭解 。並且﹐對於對稱性破壞理論中所用的真空期望值不為零做法﹐可提供數 學上的檢驗與探討。 In this dissertation, three main parts are included: (1) The first part is devoted to a new numerical method developed by us. As is well known, the difficulty resulting from the anti- commutation relation of fermion operators has been a long- standing problem in computational physics. In our work, based on the microcanonical ensemble theory in statistical mechanics, we propose and prove a new theory and then devise a method that can overcome this difficulty. In our method, we use the expectation value of the energy, as defined in quantum mechanics, instead of eigenvalue as the energy of a physical system. (2) In the second part, we improve and extend the work of Gorecki and Byers Brown. They proposed a variational theory for the ground state of a quantum system. They assumed the trial wave function in confined systems as , where acting as a variable contour function to minimize the energy. Their method gives unsatisfactory results in asymmetrical quantum problems, and therefore rarely used. We generalize their method by assuming the trial wave function as , where is a linear combination of basis functions. The meaning of our improvement is to propose a general variational method. (3) The third part was our earlier work before our direction of the research was switched into computational physics. We made a systematic survey on a simplified BCS model. We derive exact eigenfunctions of this hamiltonian. The eigenstates dispaly apparently the symmetry breaking property and the bosonic and the fermionic behaviors are seperated respectively in the ground state and in excited states. This fact is interesting in understanding the symmetry breaking phenomenon. Our work may also help to clarify the mathematical rigor of the non-zero vacuum expectation value in the symmetry breaking theory.