Abstract
We use a simple mathematical method to solve the problem of a two-dimensional hydrogen-like atom in the inhomogeneous magnetic fields B = (k/r)z and B = (k/r 3)z. We construct a Hamiltonian that takes the same form as the Hamiltonian of a hydrogen-like atom in the homogeneous magnetic fields and obtain the energy spectrum by comparing the Hamiltonians. The results show that the whole spectrum of the atom in the magnetic field B = (k/r)z can be obtained, and the problem is exactly solvable in this case. We find analytic solutions of the Schrödinger equation for the atom in the magnetic field B = (k/r 3)z for particular values of the magnetic strength k and thus present a quasi-exactly solvable model.
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 183, No. 2, pp. 329–336, May, 2015.
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Liu, L., Hao, Q. Planar hydrogen-like atom in inhomogeneous magnetic fields: Exactly or quasi-exactly solvable models. Theor Math Phys 183, 730–736 (2015). https://doi.org/10.1007/s11232-015-0291-1
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DOI: https://doi.org/10.1007/s11232-015-0291-1