B-spline方法在原子物理上的應用

Abstract

本論文研究的內容是原子物理的一部份, 屬於基礎研究。研究的主題有兩部份。 第一部份是量子橢圓態的問題。主要是計算原子(尤其是鹼金屬)的橢圓態的生命期及其他性質。眾所皆知，原子在古典極限可由行星模型來描述。然而如何能得到具橢圓分佈的量子態呢?對於純庫倫場的氫原子，物理學家利用群論的技巧，已經可以求出橢圓軌道的波函數。對於多電子系統，目前實驗室利用微波或電磁場，可以製造出芮泊原子(Rydberg atoms)的橢圓態，理論方面的相關計算卻幾乎沒有。我們已經從理論方面著手製造了芮泊原子。對不同的鹼金屬原子例如Li，Na的芮泊原子的生命期作了一個詳細的計算，同時也對芮泊原子的橢圓態，分析其生命期隨著不同橢圓離心率的變化，發現當離心率很高的時候，內核(core)的效應變得重要，與他人使用量子缺陷理論(quantum defect)得到一致的結果。 第二部份是近年來很熱門的主題-量子點的研究。 近十年來，科學的技術已能夠製造出大小約1000埃的「人工原子」，亦即它具有原子一般的特性光譜。「人工原子」將電子困在一個小空間內，其性質可由量子力學的理論來分析，所以亦稱「量子點」。量子點可以透過實驗室，技術的調整使其吸收與釋放各種波長的光。成列量子點的組合可能可以製造出無可匹敵的電腦。 「量子點」不但新鮮，而且具有實用潛力，因此吸引了許多科學家的研究。國家毫微米元件實驗室正從事相關的實驗。理論方面, 國內則有交大電物系褚德三、楊宗哲、江進福等教授，應數系劉晉良教授，彰化師大楊淳青教授已從事多年研究。大部份的理論研究都只探討非相對論的模型，我們則把它推廣到相對論性的研究。我們首先針對球形量子點模型提出系統性的討論。從非相對論到相對論的結果，以及新的現象，如能階交叉、能階平台等。接著討論量子點在外加電場情況。我們使用拋物座標，先求出量子點在無外加電場下的解析解，接著使用數值方法研究在外加電場下的拋物量子點，發現能階呈現振盪現象，這是過去沒有被提出來的。另外我們藉著調整邊界條件(例如量子點半徑、量子點位能)和外加電場的大小來控制能階交叉，以達到我們想要的輻射光源，是這個系統可能的應用。這些工作成果對於目前元件趨於量子尺寸時可能產生新的現象, 以及最新的應用, 提供一個參考。

The study in this thesis belongs to fundamental research concerning atomic physics. We focus on two major topics. The first part of this thesis is about quantum elliptical states. We calculate the lifetime and study a lot of properties of the elliptical states of atom, especially for alkaline metals. It is well known that, in classical limit, the electron orbit of an atom can be described by planet model. Can we generate a quantum state with probability of elliptical distribution? For hydrogen atom, physicists have built the wave functions of elliptical orbit using group theory. For the multi-electron systems, elliptical states of the Rydberg atoms can be produced using micro wave or electric-magnetic fields in laboratory, but theoretical calculation on these systems are rarely found. We generate Rydberg atoms numerically and calculate the lifetime of them for various alkaline atoms. We then build the elliptical states with different eccentricity. We find that when eccentricity gets large, the core effect plays an important role. This result is consistent with the other's using quantum defect theory. The second part is about quantum dots, which have attracted lots of attention recently. In the past few years, 'artificial atom' of about 1000 can be fabricated via nano-structure technology. An artificial atom is a system with electrons being confined in a small space. It has energy spectrum similar to an atom. They are also called quantum dot. A quantum dot can absorb or emit light of different wavelength in the laboratory. A supercomputer may be made of quantum dot array in future. Quantum dot is not only interesting but also of practical use. Many scientists have devoted to this topic. However, most of the publications focus on non-relativistic case. In this thesis we consider relativistic effect in quantum dot. First we study the spherical quantum dots including the non-relativistic and relativistic cases. We have found interesting properties such as level crossing and energy plateaus. Next we consider the case of a quantum dot exposed to external uniform electrical field. We use parabolic coordinates to solve the system. The wavefunction without external field is analytically solved. Its energy spectrum has many characteristics different from those of a spherical quantum dot. We also find a novel oscillation of the energy levels of the system. Moreover, due to the level crossing, we can control the radiation by adjusting boundary conditions such as the radius and potential of a quantum dot.