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Oxford University Press eBooks

The hydrogen atom

P. Anthony Cox

Published Jan 25, 1996• 0 citations
Publisher

Abstract

This chapter focuses on the hydrogen atom, which is the most important chemical application of the quantum theory. The hydrogen atom was Erwin Schrödinger's immediate goal when he developed his equation, and the solution obtained in 1926 persuaded him and indeed most interested physicists that his theory was successful. Indeed, the solutions of Schrödinger's equation — known as atomic orbitals — form the basis of our understanding, not only of atomic structure, but also of chemical bonding in molecules and solids. The chapter begins by considering the Rutherford–Bohr atom. It then looks at hydrogenic ions. The solutions described in the chapter are 'exact', in the mathematical sense that they can be obtained from the Schrödinger's equation without approximation. However, they are not quite 'correct', because this equation does not truly describe a real hydrogen atom.

DOI
10.1093/hesc/9780198559160.003.0004

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This chapter focuses on the hydrogen atom, which is the most important chemical application of the quantum theory. The hydrogen atom was Erwin Schrödinger's immediate goal when he developed his equation, and the solution obtained in 1926 persuaded him and indeed most interested physicists that his theory was successful. Indeed, the solutions of Schrödinger's equation — known as atomic orbitals — form the basis of our understanding, not only of atomic structure, but also of chemical bonding in molecules and solids. The chapter begins by considering the Rutherford–Bohr atom. It then looks at hydrogenic ions. The solutions described in the chapter are 'exact', in the mathematical sense that they can be obtained from the Schrödinger's equation without approximation. However, they are not quite 'correct', because this equation does not truly describe a real hydrogen atom.
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).
Bohr theory was not satisfactory as it failed to provide an interpretation of relative line intensities and splitting of spectral lines in the presence of electric field and magnetic field in the hydrogen spectrum. This theory also failed completely when it was applied to explain the energies and spectra of more complex atoms. In this chapter, we consider the application of Schrödinger wave theory to one-electron atom. The simplest one-electron system is the Hydrogen atom, whose Schrödinger equation can be exactly solved. Other one-electron system like He + , Li 2+ etc. can also be solved exactly. The chemical system with more than one-electron cannot be solved exactly due to electron-electron repulsion, which is the most problematic term.
Bohr theory was not satisfactory as it failed to provide an interpretation of relative line intensities and splitting of spectral lines in the presence of electric field and magnetic field in the hydrogen spectrum. This theory also failed completely when it was applied to explain the energies and spectra of more complex atoms. In this chapter, we consider the application of Schrodinger wave theory to one-electron atom. The simplest one-electron system is the Hydrogen atom, whose Schrodinger equation can be exactly solved. Other one-electron system like He + , Li 2+ etc. can also be solved exactly. The chemical system with more than one-electron cannot be solved exactly due to electron-electron repulsion, which is the most problematic term.
This chapter examines the hydrogenic atom, which is a one-electron atom or ion of general atomic number Z. It provides a set of concepts which are used to describe the structures of atoms and of molecules. It analyzes the experimental information from a study of the spectrum of atomic hydrogen and describes the results of solving the Schrödinger equation. It also considers the importance of wavefunctions to the atomic orbitals of hydrogenic atoms. The chapter talks about the atomic orbitals which are labelled by quantum numbers that specify the energy and angular momentum of an electron in a hydrogenic atom. It shows the production of energetically excited atoms when an electric discharge is passed through a gas or vapour or when an element is exposed to a hot flame.
There are certain prime features of physical systems—notably rotation and its associated angular momentum—that simply do not appear in a one-dimensional world. This chapter introduces the three-dimensional Schrodinger equation and applies it to the analysis of a particle in a three-dimensional box and to the spherically symmetric wave functions of hydrogen. The time-independent Schrodinger equation can be viewed in a way that gives it a family resemblance to many other differential equations of physics, and which expresses its structure in very general terms. A computer solution uses dimensionless variables based on natural units. The Schrodinger theory does not have the limitations of the Bohr theory. Schrodinger theory incorporates the well-established wave nature of matter and can be generalized to analyze systems for which the Bohr theory has proven inadequate: multi-electron atoms, chemical properties of atoms, molecular and solid-state physics and chemistry, as well as more subtle effects in hydrogen itself.
Introduction to Quantum Theory and Atomic Structure envelops the basic concepts needed as background for this topic, and discusses atomic structure but not molecular applications. The first two chapters are concerned with the basic ideas and problems of wave-particle duality, the nature of wavefunction, and its statistical interpretation. Chapter 3 discusses some important applications of Schrödinger's equation to chemically relevant situations. Chapters 4 and 5 deal respectively with the hydrogen atom, and with the structure of many-electron atoms and the periodic table of elements. The emphasis throughout is on the physical concepts and their concrete application.