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CRC Press eBooks

>H-Atom

Steven M. Pascal

Published Jul 12, 2023• 0 citations
Publisher

Abstract

In this chapter we solve the radial TISE for the case of a radial potential, specifically the radial potential of an electron attracted to the proton which is the nucleus of a hydrogen atom. Again, we rely on solutions to a related equation solved by a French mathematician, this time named Laguerre, whose polynomials are, in this chapter, multiplied by a simple exponential function to produce the radial wave functions. These functions contain the principle quantum number, which corresponds to the row number in the periodic table. Multiplying these radial solutions by the spherical harmonics from Chapter 14 produces the full three-dimensional solutions for the hydrogen atom, also known as the atomic orbitals. The relationship between the quantum numbers n , l , and m arise naturally from the possible combinations of radial solutions with spherical harmonics.

DOI
10.1201/9781003356172-20

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In this chapter we solve the radial TISE for the case of a radial potential, specifically the radial potential of an electron attracted to the proton which is the nucleus of a hydrogen atom. Again, we rely on solutions to a related equation solved by a French mathematician, this time named Laguerre, whose polynomials are, in this chapter, multiplied by a simple exponential function to produce the radial wave functions. These functions contain the principle quantum number, which corresponds to the row number in the periodic table. Multiplying these radial solutions by the spherical harmonics from Chapter 14 produces the full three-dimensional solutions for the hydrogen atom, also known as the atomic orbitals. The relationship between the quantum numbers n , l , and m arise naturally from the possible combinations of radial solutions with spherical harmonics.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
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).
This chapter physical properties of the hydrogen atom and its Hamiltonian function as a particle in a spherically symmetric potential having a wave function as a product of the radial function and spherical harmonic function. Hereafter, it comes to two independent wave equations: the radial equation and the angular equation. After some derivation, it is shown that the angular equation is a product of the azimuthal angle equation and polar angle equation (as discussed in the previous chapter). The azimuthal angle equation is similar to the particle in a ring problem from the previous chapter and after some derivation the polar angle equation becomes an associated Legendre equation with well-known polynomials as solutions. Next, the solution to the radial equation is presented which becomes after some derivation an associated Laguerre equation with well-known polynomials as solutions and from this result is obtained the energy equation for the hydrogen-like atom. Subsequently, it is derived the normalization constant of the radial function. Afterwards, the numerical analysis of the radial equation of the hydrogen atom along with its corresponding Fortran code is presented. Finally, the hydrogen atomic orbitals equations and plots are depicted followed by the fine structure of the hydrogen (198 words).
With some understanding of the TISE in one dimension behind us, it is now time to graduate to the three-dimensional TISE. This will be required if we wish to understand electrons and atoms, which reside in three-dimensional space. The best coordinate system to use to study atoms is the spherical coordinate system (r, θ, ϕ), so we will only use this system. For a potential that is radial (that is, a central potential, such as an electron drawn to the center of the atom by the nucleus), the solutions ψ(r, θ, ϕ) can be split into separate functions of r, θ, and ϕ, similar to how Ψ(x,t) was separated into ψ(x) times f(t). Thus, the 3D TISE can be split into three separate TISE, one for each dimension r, θ, and ϕ. Since we are considering only radial potentials V(r) with no θ and ϕ dependence, the θ and ϕ TISE do not change. We provide the solutions for these two equations in this chapter, and needn't solve them again for other radial potentials. The θ and ϕ solutions, multiplied by each other, are called the spherical harmonics.
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.