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Cambridge University Press eBooks

The hydrogen atom

Carson L. Tang

Published Jun 23, 2005• 0 citations

Abstract

The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.

DOI: 10.1017/cbo9781139165273.007

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The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.

The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.

The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge −e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the center of mass coordinate and the relative coordinates. In the hydrogen atom the attractive force exerted by the proton on electron prevents it from escape. The potential energy V of the system is −e2/r where r is the distance between the electron and the proton. Suppose the nucleus is held fixed at the origin of coordinates. V (r) is independent of θ and φ, that is, insensitive to the direction of r and depends only on the scalar magnitude r. The potential energy remains invariant under rotation and is said to be spherically symmetric. Figure 12.1 shows V (r) as a function of r. In this chapter the hydrogen atom problem in threedimension is solved and momentum eigenfunction is obtained. The problem in D-dimension where D may be noninteger is analysed. The magnetic field produced by a hydrogen atom is worked out. Finally, the system in parabolic coordinates is discussed.

The world should have waited until 1913, when a Danish physicist N. Bohr proposed an entirely new theory for a description and partial explanation of the spectral properties of the hydrogen atom. His theory was based on two postulates, the so-called Bohr postulates, and both of them were impossible to be understood from the point of view of classical physics. Being based on these postulates and employing classical physics, N. Bohr was able to derive explicit expressions for the energy levels of the hydrogen atom and the wavelengths of possible transitions. The hydrogen-like atom has only one electron, and all others have been removed by various ways of ionization. An essential feature of these systems is the presence of one electron only. The Coulomb interaction between this electron and the nucleus is spherically symmetrical. Later, we shall see that the situation becomes considerably more complicated in the case of multielectron atoms.

In 1897, Thomson demonstrated that cathode rays (observed to be emitted from the cathodes of vacuum tubes) were in fact charged particles which he called electrons. Thomson proposed that the atom consisted of a positively charged sphere in which were embedded negatively charged electrons. Rutherford subsequently found in 1911 that the electrons orbited at some distance from a central positively charged nucleus. Rutherford proposed that electrostatic attraction between the nucleus and the electron was balanced by the centrifugal force arising from the orbital motion. However, if this were the case, then the electrons (being accelerated inwards towards the center of rotation) would continuously radiate all their energy as electromagnetic waves and very quickly fall into the nucleus. In 1913, Bohr postulated two important additions to Rutherford’s theory of atomic structure: 1. Electrons can orbit the nucleus in what are called stationary states in which no emission of radiation occurs and in which the angular momentum L is constrained to have values: 2. Electrons can make transitions from one state to another accompanied by the emission or absorption of a single photon of energy E = hf thus leading to absorption and emission spectra.

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 illustrates Niels Bohr's ideas in some detail. In the formulation of his model, Bohr assumed that the electron in the hydrogen atom moves under the influence of the Coulomb attraction between it and the positively charged nucleus, as assumed in classical mechanics. The Bohr model was very successful in explaining the discrete atomic spectra of one-electron atoms. The Bohr model is, of course, a theoretical model of an atom. Using Bohr's quantization postulate, find the linear velocity of the electron's sphere. To find the radius of the orbit, Bohr postulated that the angular momentum of the electron is quantized, i.e., it can only take values that are integer multiples of ħ. The chapter begins with considering the classical equation of motion for the electron in a circular orbit, which is based on Newton's laws of motion and Coulomb's law of electric force.