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研究生:王詩雅
研究生(外文):Shiya Wang
論文名稱:強磁場中的氫原子結構--一個二體分布的問題
論文名稱(外文):Hydrogen Atomic Structure in Strong Magnetic Fields -- a two-particle approach
指導教授:張祥光張祥光引用關係
指導教授(外文):Hsiang Kuang Chang
學位類別:碩士
校院名稱:國立清華大學
系所名稱:物理學系
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:50
中文關鍵詞:強磁場氫原子
外文關鍵詞:Strong Magnetic FieldsHydrogen Atomic Structure
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隨著強磁場在天體上的發現,例如:表面大約有10^6 〜 10^9高斯 (Gauss) 的白矮星 (white dwarfs)、表面大約有10^9 〜 10^13高斯的中子星 (neutron stars)、 以及一般認為表面具有超強磁場約10^14 〜 10^15高斯的緻密星體 (magnetars),計算此環境之下的原子能階因而在高能天文物理上佔有舉足輕重的地位。特別值得注意的是,強磁場對於原子結構、能階 (energy levels) 等性質,產生了極為重大的改變。中子星表面的原子結構尤其令我們感到興趣,因為缺少此項計算的大氣模型,將無法提供符合實際情形的束縛─束縛 (bound-bound) 與束縛─自由 (bound-free) 反應下的不透明度 (opacities),以進而計算光譜。本論文中,我們以二體 (two-particle) 系統來處理一個強磁場約10^12高斯的氫原子。在簡諧近似 (simple harmonic approximation) 的假設之下,我們選用圓柱座標下的B-splines作為波函數 (wave functions) 的基底 (basis functions) 。
和其他作者不同的做法是,本論文由下列兩個觀點出發:(1) 強磁場中,質子的分布 (the proton spreading) 不但不應該忽略,反而會導致計算所得的束縛能(binding energy) 產生明顯的改變;(2) 強磁場中,粒子的質心運動 (center-of-mass motion) 與相對運動 (relative motion),以量子力學的觀點,不能夠被分離。由此,我們計算出氫原子低能階的能量。與其他作者已發表的結果比較之後發現,我們得到較大的束縛能,此正顯示庫倫作用力 (Coulomb interaction) 與電子的藍道能階能量 (Landau level energy) 相比之下,並不似一般認為的小,反而是可與之比擬的。然而,我們所得只是初步,仍然存在些問題有待未來釐清並加以改進。
With the discovery of extremely high magnetic fields of about 10^6 ~ 10^9 Gauss for white dwarfs, 10^9 ~ 10^13 Gauss for neutron stars, and probably 10^14 ~ 10^15 Gauss for magnetars, the calculation of energy levels of atoms is very important for high energy astrophysics. Particularly, strong magnetic fields of neutron stars substantially modify atomic structures, energy levels, and other properties. Without calculating the structures of atoms on the surface of neutron stars, opacities due to bound-bound and bound-free processes cannot be acquired, and computed spectra are rather unrealistic. In our work, we treat a two-body problem of a hydrogen atom with a simple harmonic approximation in strong magnetic fields about 10^12, using B-splines as the basis functions of wave functions in cylindrical coordinates.
Different from previous works by other authors, we start from the following two considerations: (1) the proton spreading due to strong magnetic fields is not negligible, and it leads to a significant change of the computed binding energy; (2) the center-of-mass motion and the relative motion of particles in strong magnetic fields can not be separated in a quantum mechanical view. We calculate the energy levels of low-lying states, and compare these results with other published ones. We obtain remarkable difference from others— the binding energy is much larger than others and shows that the Coulomb interaction is almost comparable with the landau level energies of electrons. However, our work is only a preliminary one. There still exist some issues to be clarified and improved in the future.

1. Introduction . . . . . . . . . . . . . . . . . . . . . . .1
2. Method . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1. Hamiltonian of a hydrogen atom in strong magnetic
fields . . . . . . . . . . . . . . . . . . . . . . . . .4
2.2. Wave functions . . . . . . . . . . . . . . . . . . . .6
2.2.1. Landau levels . . . . . . . . . . . . . . . . . .6
2.2.2. Simple harmonic approximation . . . . . . . . . . 7
2.3. The eigensystem . . . . . . . . . . . . . . . . . . . 8
3. A brief introduction to B-splines . . . . . . . . . . . 10
3.1. Divided differences . . . .. . . . . . . . . . . . . 11
3.2. Definition of B-splines . . . . . . . . . . .. . . . 12
3.3. Representation of B-splines . . . . . . . . . . . . 14
4. Numerical calculation . . . . . . . . . . . . . . . . . .17
4.1 Eigenvectors . . . . . .. . . . . . . . . . . . . . . 18
4.1.1 rho_e-part . . . . . . . . . . . . . . . . . . . 18
4.1.2 eho_p , z -part . . . . . . . . . . . . . . . . . 18
4.2 Potential energy . . . . . . . . . . . . . . . . . . 21
4.3 Computing time and degree of precision . . . .. . . . 25
5. Results . . . . . . . . . . . . . . . . . . . . . . . . .27
5.1. Eigenvalues . . . .. . . . . . . . . . . . . . . . . 27
5.2. Error discussion . . . . . . . . . . . . . . . . . . 29
5.2.1. error from integration . . . . . . . . . . . . . 29
5.2.2. convergence . . . . . . . . . . . . . . . . . . 29
5.3. Eigenvectors . . . . . . . . . . . . . . . . . . . . 35
6. Discussions . . . . . . . . . . . . . . . . . . . . . . .44
6.1. Comparison with other published results . . . .. . . 44
6.2. Implication of our results . . . . . . . . . . . . . 46
7. Summary . . . . . . . . . . . . . . . . . . . . . . . . .48

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