High-accuracy numerical calculations of the bound states of a hydrogen atom in a constant magnetic field with arbitrary strength

We develop a simple and effective method for solving the Schrödinger equation of a hydrogen atom in a constant magnetic field with arbitrary strength. Energies are obtained not only for the ground and low-lying states but also for highly excited states with precision from 12 up to 20 decimal digits. The calculations are performed for an entire range of magnetic field intensity up to 9.4 x 10^8 Tesla, the strongest field ever observed. The strong point of the development of the method is the construction of an anharmonic oscillator model for a hydrogen atom in a constant magnetic field via the Kustaanheimo–Stiefel transformation. This model allows the use of purely algebraic calculations and the Feranchuk–Komarov (FK) operator method for effectively solving the Schrödinger equation. The advantages of the basis set in this work are also discussed to extend its application to other problems, such as multi-electron atoms in a constant magnetic field. We also provide a program written by FORTRAN for the solutions mentioned above.

本研究提出一种简洁高效的方法，用于求解任意强度恒定磁场中氢原子的薛定谔方程（Schrödinger equation）。不仅可获取基态与低激发态的能级，还能以12至20位十进制的精度得到高激发态的能级结果。计算覆盖了磁场强度的全部取值范围，最高可达9.4×10^8特斯拉（Tesla）——这是目前已观测到的最强磁场。该方法的核心亮点在于，通过库斯塔尼莫-斯蒂费尔变换（Kustaanheimo–Stiefel transformation）构建了恒定磁场中氢原子的非简谐振子模型：该模型可采用纯代数运算，并借助费兰丘克-科马罗夫（Feranchuk–Komarov，FK）算子方法高效求解薛定谔方程。本研究还讨论了本次工作所采用基组的优势，以将该方法推广至其他相关问题，例如恒定磁场中的多电子原子。此外，我们还为上述求解任务提供了Fortran语言编写的程序。