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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Schrodinger equation solution for the hydrogen atom with consideration of nucleus rotation (spin)

В. И. Башков, D. Zenkin

Published Jan 14, 2000• 0 citations
Publisher

Abstract

The variant of the Schroedinger equation solution for the atom of hydrogen is submitted in view of the nucleus rotation, i.t. with the account received earlier by one of the authors potential energy for a rotating pointed charged particle. The received result allows by a quasi-classical way to take into account in the analytical decision of the Schroedinger equation for atom of hydrogen additional splitting of the atom energy levels because of rotation of the atomic nucleus.

DOI
10.1117/12.375352

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The variant of the Schroedinger equation solution for the atom of hydrogen is submitted in view of the nucleus rotation, i.t. with the account received earlier by one of the authors potential energy for a rotating pointed charged particle. The received result allows by a quasi-classical way to take into account in the analytical decision of the Schroedinger equation for atom of hydrogen additional splitting of the atom energy levels because of rotation of the atomic nucleus.
Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.
In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r−l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally. 1.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
The Born and Jordan matrices representing the "coordinates" of the hydrogen atom are calculated. The model is that of an electron revolving without rotation about a positive nucleus. The computation is based on a method developed independently by the author and by E. Schroedinger. Half quantum numbers appear in the formulas for the matrices. It is stated without proof, that they also enter into the formula for the relativistic fine structure. This will necessitate the consideration of a more elaborate model for the hydrogen atom (the rotating electron of Uhlenbeck and Goudsmit). The intensity ratio of the two brighter components of ${\mathrm{H}}_{\ensuremath{\alpha}}$ is calculated and found to be 10:2.1 in contradiction with the observed 10:8. For ${\mathrm{H}}_{\ensuremath{\beta}}$ the calculated ratio is 10:3.6, and the observed 10:9. The discrepancy is too great to be accounted for by the incompleteness of the model, and points to a difficulty in the Born and Jordan theory.
The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.