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College Physics

Treating the ground states of hydrogen, positronium and helium atoms by Bohr model uniquely

WU Qu-fei

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

By means of the method, that dealt with the ground states of hydrogen and positronium using the Bohr model, a classical model of the helium ground state is assumed and its energy and radius are calculated. The result is compared with experiment and variational method of quantum mechanics.

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The Bohr model is historically important in the understanding of quantum physics. It gives many important results for the hydrogen atom including atomic size and ionization energy. It is sometimes referred to as a semi-classical calculation, because it includes the physics of angular momentum quantization without the full quantum mechanical description of an electron wave. It is tremendously useful for gaining physical intuition of the quantum nature of the atom. In the Bohr model of the hydrogen atom, an electron has a circular orbit with speed v and radius r about the proton. The Bohr quantization condition is that an integer number of electron (de Broglie) wavelengths must fit to the atomic circumference.
In this paper, we accurately estimate the ground-state energy and the atomic radius of the helium atom and a helium-like Hookean atom by employing the uncertainty principle in conjunction with the variational approach. We show that with the use of the uncertainty principle, electrons are found to be spread over a radial region, giving an electron cloud. Our calculation also shows how the Coulomb interaction between electrons affects their distribution. This leads to a physical picture of how electrons are located with respect to each other in these atoms. Finally, we also obtain through our calculations a general formula for the estimate of ground-state energy and radius of two electron atoms and ions with atomic number Z.
The Bohr model is highly successful in explaining many features of the hydrogen atom. The ground state wave function does not have any angular dependence. The normalized hydrogen atom wave functions are directly accessible in Mathematica and can be readily verified as solutions of the Schrodinger equation. Get the hydrogen atom ground state wave function in terms of the Bohr radius a. The probability of finding the electron per volume at a given location is given by the square of the wave function. The electron has only a very tiny probability of being found inside the proton, whose size is about 1 fm.