Calculation of the energy levels of a hydrogen atom in a magnetic field of arbitrary strength by usingBsplines

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Abstract
B splines with carefully adjusted knot sequences are used as basis functions in cylindrical coordinates to calculate the energy levels of the low-lying (m=0) states of a hydrogen atom in a uniform magnetic field of arbitrary strength by using the variational method. The strength of the field calculated covers a range from γ=0 to γ=100 000 for the ground state and up to γ=2000 for the 2s and 2p states. A precision of up to ten digits in the region of 0≤γ≤200 and up to eight digits for γ≳200 has been maintained for all the results presented. In order to test the applicability of the B splines for the higher excited states, the energy levels of 3s, 3p, 3d, and 4f states with m=0 in the field region of 0≤γ≤10 have also been calculated. Even though the number of basis sets was kept constant, the accuracy of the results was maintained in the intermediate field regions and also in the very strong field regions, which are known to be difficult for achieving high accuracy. The flexibility of the B splines is demonstrated here by the uniformly accurate results in the transition from spherical symmetry to Landau symmetry as the field strength of the magnetic field is increased. The energy levels of all the low-lying states with various ranges of magnetic field strength have been compared with published values. They are in good agreement with the most updated values and compare favorably with all other published results.

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