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Cambridge University Press eBooks

THE ATOM

B. S. Chandrasekhar

Published Oct 16, 1997• 0 citations
Publisher

Abstract

Introduction I give in this chapter the quantum-mechanical description of an atom. I look in some detail at the hydrogen atom, which is the simplest atom we have, with just one proton and one electron. From the picture that emerges, we shall be able to see how to describe all other atoms. Before continuing, I need to explain what I mean when I say that the proton and electron have equal and opposite electric charges. It is because of these charges that the atom is stable, and does not spontaneously break up into the two separate particles. Electric charge and potential energy If I take a ball in my hand and then let go, it drops to the floor. This is because of the gravitational interaction between the ball and the earth. The ball had gravitational potential energy when I held it, and when I release it, this energy is progressively converted to kinetic energy as it falls. Gravitation makes two bodies move towards each other, and not away from each other: the gravitational force is always attractive. As a result of many experiments done over the years, we find that there is another kind of force in nature, somewhat analogous to, but in one aspect different from, gravitation. I take two idealized examples, and use them to show the features of this force. First I consider two electrons that I imagine to be pinned down a distance r apart.

DOI
10.1017/cbo9780511524202.005

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Introduction I give in this chapter the quantum-mechanical description of an atom. I look in some detail at the hydrogen atom, which is the simplest atom we have, with just one proton and one electron. From the picture that emerges, we shall be able to see how to describe all other atoms. Before continuing, I need to explain what I mean when I say that the proton and electron have equal and opposite electric charges. It is because of these charges that the atom is stable, and does not spontaneously break up into the two separate particles. Electric charge and potential energy If I take a ball in my hand and then let go, it drops to the floor. This is because of the gravitational interaction between the ball and the earth. The ball had gravitational potential energy when I held it, and when I release it, this energy is progressively converted to kinetic energy as it falls. Gravitation makes two bodies move towards each other, and not away from each other: the gravitational force is always attractive. As a result of many experiments done over the years, we find that there is another kind of force in nature, somewhat analogous to, but in one aspect different from, gravitation. I take two idealized examples, and use them to show the features of this force. First I consider two electrons that I imagine to be pinned down a distance r apart.
Some preliminaries We have, until now, thought of atoms as tiny hard spheres that feel a force of attraction towards one another. We ignored the fact that an atom consists of a nucleus surrounded by one or more electrons. If we stopped at this, we would make little progress towards understanding the properties of solids, other than that the atoms may form crystals. We must therefore go on and take a look at the internal structure of the atom itself, and see how this changes when the atom is surrounded by other atoms in a crystal. We shall then find that we need a description that is somewhat removed from our everyday experience. There is a name for this description, quantum mechanics : mechanics, because it deals with the motion of objects, and quantum, because features of the motion, such as energy, are quantized , meaning that they can only have any one of a set of distinct values, and no other value in between. Imagine for example a car which is so constructed that it can travel only at speeds of 10, or 50, or 100 km per hour, and not at any other speed. Then we could say that the speed of the car is quantized. This idea of quantization is something alien to our usual way of thinking. After all, a car is able to go at any speed we like up to its maximum speed.
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 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.
An atom consists of a nucleus, containing protons and neutrons, with various electrons swarming around the outskirts. The electrons are held in the vicinity of the nucleus as they are negatively charged and hence attracted by the positively charged protons. Although atoms are complicated objects, there are certain situations where they can be treated as simple lumps of matter. Their internal structure has no relevance to the details of what is happening. This sort of situation is exemplified by experiments in which atoms are deliberately smashed into one another at comparatively low energy and as such provide an ideal start in a study of the quantum physics of identical particles. When we are dealing with atoms, there is something more to being ‘identical’ than we can appreciate with larger-scale objects. Atoms of the same type are really identical, measurably identical if you like, which can make a considerable difference in some physically important situations.
Anatomy of an atom In the second half of my story, the three strands of relativity, quantization and symmetry will be braided together. Before we embark on this journey, let us take a rest and consider some ways in which the properties of the small-scale quantum world govern our world at large. Most people wonder about the relevance of subatomic studies for our understanding of ourselves. Quite a few popular accounts of particle physics contain a heavy dose of reflections upon Larger Issues and related Deep Thoughts. I will try to avoid that, but it cannot be denied that the connections between the subatomic scale and our daily world are strong and direct, albeit invisible to our large-scale senses. Let us first consider atoms, the building blocks that have the most direct significance in our daily lives (in the remaining chapters we will discuss the structure of the particles of which atoms are made). For simplicity and definiteness, and because we have encountered it in previous discussions, I will mainly speak of the hydrogen atom. Its nucleus is a proton, having one unit of positive electric charge. A neutral hydrogen atom is formed when an electron, bearing one negative unit of electric charge, binds to it. The binding makes the atom electrically neutral.
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 centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.