With the exception of the latter half of chapter 6 we have, up to this point, been discussing single-particle quantum mechanics. This is easy (although you may not think so). One can certainly gain a lot of insight and understanding from a single-particle theory. However, in real life there are very few (no) situations in physics in which a single-particle theory is able to paint the whole picture. Any real physical process involves the interaction of many particles. In fact even that is a vast simplification. Really any physical process involves the interaction of all particles. Even the gravitational attraction due to an electron at the other end of the universe is felt by an electron on earth. To describe all particles in a calculation is, of course, absurd. You would have to include the particles of the paper you are writing the calculation down on and the particles in your brain thinking about the many-body problem. However, many-body theory on a more limited scale is feasible. In this chapter we discuss two ways of going beyond the one-electron approximation. Actually, we are not going very far beyond the one-electron approximation and you will see what is meant by that soon. This chapter is essentially divided into two halves.
With the exception of the latter half of chapter 6 we have, up to this point, been discussing single-particle quantum mechanics. This is easy (although you may not think so). One can certainly gain a lot of insight and understanding from a single-particle theory. However, in real life there are very few (no) situations in physics in which a single-particle theory is able to paint the whole picture. Any real physical process involves the interaction of many particles. In fact even that is a vast simplification. Really any physical process involves the interaction of all particles. Even the gravitational attraction due to an electron at the other end of the universe is felt by an electron on earth. To describe all particles in a calculation is, of course, absurd. You would have to include the particles of the paper you are writing the calculation down on and the particles in your brain thinking about the many-body problem. However, many-body theory on a more limited scale is feasible. In this chapter we discuss two ways of going beyond the one-electron approximation. Actually, we are not going very far beyond the one-electron approximation and you will see what is meant by that soon. This chapter is essentially divided into two halves.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are signicant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benets of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantum-mechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic eects become observable, for which non-relativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.33) must be used. There are, however, more subtle eects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are significant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benefits of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantummechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic effects become observable, for which nonrelativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.34) must be used. There are, however, more subtle effects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
The calculation of a wavefunction took about two afternoons, and five wavefunctions were calculated in the whole …. Wigner and Seitz, 1933 Summary In order to explain many important properties of materials and phenomena, it is necessary to go beyond independent-particle approximations and directly account for many-body effects that result from electronic interaction. The many-body problem is a major scientific challenge, but there has been great progress resulting from theoretical developments and advances in computation. This chapter is a short introduction to the interacting-electron problem, with some of the history that has led up to the concepts and methods described in this book. The many-body interacting-electron problem ranks among the most fascinating and fruitful areas of research in physics and chemistry. It has a rich history, starting from the early days of quantum mechanics and continuing with new intellectual challenges and opportunities. The vitality of electronic structure theory arises in large part from the close connection with experiment and applications. It is spurred on by new discoveries and advances in techniques that probe the behavior of electrons in depth. In turn, theoretical concepts and calculations can now make predictions that suggest new experiments, as well as provide quantitative information that is difficult or not yet possible to measure experimentally. This book is concerned with the effects of interactions between electrons beyond independent-particle approximations. Some phenomena cannot be explained by any independent-electron method, such as broadening and lifetime of excited states and two-particle bound states (excitons) that are crucial for optical properties of materials. There are many other examples, such as the van derWaals interaction between neutral molecules that arises from the dipole–induced dipole interaction. This force, which is entirely due to correlation between electrons, is an essential mechanism determining the functions of biological systems. Other properties, such as thermodynamically stable magnetic phases, would not exist if there were no interactions between electrons; even though mean-field approximations can describe average effects, they do not account for fluctuations around the average. Ground-state properties, such as the equilibrium structures of molecules and solids, can be described by density functional theory (DFT) and the Kohn–Sham independent-particle equations. However, present approximations are often not sufficient, and for many properties the equations, when used in a straightforward way, do not give a proper description, even in principle.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
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.
Those defile the purity of mathematical and philosophical truthy who confound real quantities with their relations and sensible measures. —ISAAC NEWTON, Principia Before we say anything about quantum mechanics, let us take a quick look at the Newtonian (or ‘classical’) view of the physical universe. According to that view, nature consists of an enormous number of particles. When Newtonian physics is combined with the theory of the electromagnetic field, it becomes convenient to think of these particles as dimensionless (even if there is a kind of conceptual strain involved in trying to think of something as having a mass but not any size ), and as possessing electrical properties – negative charge, or positive charge, or neutrality. This leads to the well-known ‘solar system’ view of the atom – with the electrons whirling around the nucleons (neutrons and protons) just as the planets whirl around the sun. Out of atoms are built molecules; out of molecules, macroscopic objects, scaling from dust motes to whole planets and stars. These latter also fall into larger groupings – solar systems and galaxies – but these larger structures differ from the ones previously mentioned in being held together exclusively by gravitational forces. At every level, however, one has trajectories (ultimately that means the possibility of continuously tracing the movements of the elementary particles) and one has causality (ultimately that means the possibility of extrapolating from the history of the universe up to a given time to its whole sequence of future states).