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IOP Conference Series Materials Science and Engineering

The rule of radius averaging in hydrogen atom

Abdurrouf

Published Jun 1, 2019• 0 citations
Publisher

Abstract

Abstract According to quantum mechanics, electrons do not have a fixed position in an atom, and therefore orbitals have no definite radii. However, electrons have characteristic wave functions from which the radius of their orbits can be calculated or averaged. Depending on the average method, there are three most popular expressions for orbital radius, namely the average radius, the root mean square (rms) radius, and the most probable radius. Unfortunately, for hydrogen atom, none of those three radii is equal to the classical radius, even for large principal quantum numbers called the classical regime. Here, by using energy analysis, we propose a harmonic radius and show that the results well agree with the classical radius for each principal and orbital quantum numbers.

DOI
10.1088/1757-899x/546/5/052002

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Abstract According to quantum mechanics, electrons do not have a fixed position in an atom, and therefore orbitals have no definite radii. However, electrons have characteristic wave functions from which the radius of their orbits can be calculated or averaged. Depending on the average method, there are three most popular expressions for orbital radius, namely the average radius, the root mean square (rms) radius, and the most probable radius. Unfortunately, for hydrogen atom, none of those three radii is equal to the classical radius, even for large principal quantum numbers called the classical regime. Here, by using energy analysis, we propose a harmonic radius and show that the results well agree with the classical radius for each principal and orbital quantum numbers.
The electronic transition energies from the electron clouds are the same as predicted by the Bohr model. The hydrogen atom, most probable radius of the first quantum mechanical electron cloud is identical to the radius of the first Bohr orbit. The quantum mechanical description goes much further, however. Not only does it predict a series of increasingly larger spherical electron clouds, but other electron cloud series as well. Because of their three dimensional shapes, these electron clouds sometimes are said to describe space states. The analogy to planetary orbits ends when we grasp the fact that quantum mechanical electron clouds are three-dimensional, where a Bohr orbit would be a circle lying entirely in a plane. Quantum mechanics also predicts the ionization energy of the hydrogen atom.
The simplest atom consisting of a proton and an electron is a hydrogen atom. The energy level and the probability of finding an electron in a hydrogen atom can be determined based on the function of radial wave. The function of radial wave of a hydrogen atom is influenced by the principal quantum number ( n ) and the orbital quantum number ( l ). Completion of the function of radial wave of a hydrogen atom in the principal quantum numbers( n ) 4 and 5 uses the time-independent Schrodinger equation approach in spherical coordinates, variable separation method and uses the associated Laguerre function. The normalized of the function of radial wave of a hydrogen atom R nl ( r ) at n = 4 produces four wave functions:R 40 ( r ); R 41 ( r ); R 42 ( r ); R 43 ( r ), whereas at n = 5 produces five wave functions:R 50 ( r ); R 51 ( r ); R 52 ( r ); R 53 ( r ); R 54 ( r ). The results of this research in the form of the functions of radial wave of hydrogen atom can be used to determine the behavior of electron in a hydrogen atom.
The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})$, in which PPP, $\ensuremath{\Theta}$, and $\ensuremath{\Phi}$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates rrr, $\ensuremath{\theta}$, and $\ensuremath{\varphi}$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})=\left\{\frac{1}{{(2\ensuremath{\pi})}^{\frac{1}{2}}}{e}^{\ifmmode\pm\else\textpm\fi{}\mathrm{im}\ensuremath{\Phi}}\right\} \left\{{\left(\frac{(2l+1)(l\ensuremath{-}m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{{P}_{l}}^{m}(cos\ensuremath{\Theta})\right\}$ $\left\{\frac{\ensuremath{\pi}{2}^{2l+4}l!}{{(\ensuremath{\gamma}h)}^{\frac{3}{2}}}{\left(\frac{n(n\ensuremath{-}l\ensuremath{-}1)!}{(n+l)!}\right)}^{\frac{1}{2}}\frac{{\ensuremath{\zeta}}^{l}}{{({\ensuremath{\zeta}}^{2}+1)}^{l+2}}{C}_{n\ensuremath{-}l\ensuremath{-}1}^{l+1}\left(\frac{{\ensuremath{\zeta}}^{2}\ensuremath{-}1}{{\ensuremath{\zeta}}^{2}+1}\right)\right\}$ in which $\ensuremath{\zeta}=(\frac{2\ensuremath{\pi}}{\ensuremath{\gamma}h})P$, with $\ensuremath{\gamma}=(\frac{4{\ensuremath{\pi}}^{2}\ensuremath{\mu}{e}^{2}Z}{n{h}^{2}})=(\frac{Z}{n{a}_{0}})$. The probability ${\ensuremath{\Xi}}_{\mathrm{nl}}(P)\mathrm{dP}$ that the electron have a total momentum lying within the limits PPP and P+dPP+dPP+dP is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.
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.
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).