单电子系统塞曼效应的精确解
Exact solution of the Zeeman effect in single-electron systems
摘要：
与普遍的看法相反，塞曼效应可以在单电子系统中被精确地处理，对于任意的磁场强度，只要磁场中的二次项可以被忽略。这些公式实际上早在1927年左右就由达尔文利用角动量的经典图像推导出来了，并在1933年由贝特以适当的量子力学形式提出，尽管没有任何证明。这些表达式已经或多或少地从文献中消失了;相反，现在的传统处理方法是分别只给出弱场和强场的近似。然而，在聚变研究和其他等离子体物理应用中，用于控制等离子体形状和位置的磁场跨越了从弱到强的整个区域，需要统一处理。本文用量子力学方法详细推导了静态磁场中类氢原子和类氢离子的本征能和本征态。值得注意的是，这些公式并不比更著名的近似复杂得多。此外，该推导允许电子自旋旋磁比$g_s$的值不同于2。为了完整起见，我们回顾了两个类氢能级之间偶极跃迁的细节，并计算了相应的塞曼谱。文中还详细讨论了在推导过程中所作的各种近似。
英文摘要：
Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum-mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position of the plasma span the entire region from weak to strong fields, and there is a need for a unified treatment. In this paper we present the detailed quantum-mechanical derivation of the exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static magnetic field. Notably, these formulas are not much more complicated than the better-known approximations. Moreover, the derivation allows the value of the electron spin gyromagnetic ratio $g_s$ to be different from 2. For completeness, we then review the details of dipole transitions between two hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various approximations made in the derivation are also discussed in details.
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