Lie algebraic approach to the Hellmann Hamiltonian by considering perturbation method

We show that the Lie algebraic approach with the perturbation method can be used to study the eigenvalues of the Hellmann Hamiltonian. The key element is the Runge–Lenz vector, which appears in problems with radial symmetry. This symmetry implies that the proper lie algebra for these Hamiltonians is $so(4)$, which is a sum of two $so(3)$ Lie algebras and requires symmetry of the angular momentum vector $\vec{L}$ and the Runge–Lenz vector $\vec{M}$, and therefore their cross products as $\vec{W}=\vec{L}\times\vec{M}$. Here, Yukawa potential is considered as a perturbation term, which is added to the Coulomb Hamiltonian to produce the Hellmann Hamiltonian. Lie algebraically, the perturbation term adds a magnitude of precession rate $\Omega$ to all three operators $\vec{L}$, $\vec{M}$, and $\vec{W}$. Topologically, we show that the appearance of this precession has a significant effect on the spectrum and the corresponding Lie algebra of the Hellmann potential. By using Lie algebraic properties of the Runge–Lenz vector and using the Kolmogorov method, we obtain the energy spectrum of this Hamiltonian.