Hydrogen Atom

  • Alessandro Teta
Part of the UNITEXT for Physics book series (UNITEXTPH)


We study the dynamics of a particle subject to an attractive central force of Coulomb type as a model for the hydrogen atom. In the first part of the chapter, we use qualitative methods to discuss self-adjointness and lower boundedness of the Hamiltonian, to characterize the essential spectrum, to prove the absence of positive eigenvalues and the existence of infinitely many negative eigenvalues accumulating at zero, and to estimate the lowest eigenvalue. Starting from Sect. 9.6, we approach the explicit solution of the eigenvalue problem for the Hamiltonian using spherical coordinates and exploiting the conservation of the angular momentum. We first compute the eigenvalues of the square \(L^2\) and the third component \(L_3\) of the angular momentum and determine a system of common eigenvectors, the spherical harmonics. Using this fact, we are reduced to study an ordinary differential equation for a function of the radial variable. Solving the equation, we compute the eigenvalues and the eigenvectors of the Hamiltonian.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica Guido CastelnuovoUniversità degli Studi di Roma “La Sapienza”RomeItaly

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