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As the simplest atom in nature, the hydrogen atom has been explored thoroughly from the perspective of non-relativistic quantum mechanics to relativistic quantum mechanics. Among the research on hydrogen atom, its energy level is the most basic, which can be obtained more conveniently predicated on the SO (4) symmetry than the wave-equation resolution. Moreover, ``spin'' is another indispensable topic in quantum mechanics, appearing as an intrinsic degree of freedom. In this work, we generalize the quantum Runge-Lenz vector to a spin-dependent one, and then extract a novel Hamiltonian of hydrogen atom with spin based on the requirement of SO (4) symmetry. Furthermore, the energy spectrum of hydrogen atom with spin potentials is also determined by the remarkable approach of SO (4) symmetry. Our findings extend the ground of hydrogen atom, and may contribute to other complicated models based on hydrogen atom.
Fan et al. (Wed,) studied this question.