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A unified account of the field rather than a description of parallel methods, the text's main thematic approaches include the self-consistent field and its generalizations, perturbation theory and the use of Feynman diagrams, and the use of Green functions to describe excitations of a many-body system. The primary emphasis is on the theories of atomic nuclei, the electron gas, superconductivity, and liquid helium three. A familiarity with the principles of nonrelativistic quantum mechanics and statistical mechanics is assumed, but a detailed knowledge of nuclear and solid state physics is unnecessary.
Product Details
| ISBN-13: | 9780486782850 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 12/03/2013 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 256 |
| File size: | 12 MB |
| Note: | This product may take a few minutes to download. |
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CHAPTER 1
Introduction
The many-body systems which we consider in this book are met in several different physical contexts, and so the ideas presented here can be applied to problems which arise in various branches of physics. In nuclear physics, atomic physics, solid-state physics, and low-temperature physics, the types of problem which we discuss arise frequently. It is not so obvious that they also occur in elementary particle physics, but the fact that ideas developed for the study of elementary particles have been applied to many-body problems is an indication that they do indeed occur.
We discuss only those systems for which the symmetry or antisymmetry of the wave functions with respect to interchange of the particles has a dominating influence on the properties of the system. There are, of course, other many-body systems whose properties can only be explained in terms of quantum mechanics; a gas of diatomic molecules is such a system. For such a system, quantum theory is needed only to explain the properties of the individual molecules, and the many-body problem can then be solved without further reference to quantum mechanics. The properties of the systems with which we are concerned here can only be understood in terms of the quantum theory of many-body systems. It is not possible to regard the understanding of the quantum effects and of the many-body properties as two distinct problems. With such systems, it is of the utmost importance whether the constituent particles are bosons, so that the wave function is symmetric, or fermions, so that the wave function is antisymmetric.
The best examples of this sort of system are the two isotopic forms of liquid helium. Both appear to be liquids right down to zero temperature, and this is a fact which cannot be explained by classical mechanics. There is a striking difference between the properties of the two isotopes, much greater than would be expected from the difference in mass between them. Liquid He4 is superfluid below 2.2 °K, flowing in narrow tubes with no apparent viscosity, while liquid He3 behaves like a normal fluid down to less than 0.003°K. This difference is almost certainly due to the fact that the atoms of He4 are bosons, while the atoms of He3 are fermions.
Apart from liquid He3, there are several other examples of many-fermion systems whose properties are strongly affected by the antisymmetry of the wave functions. The electrons in an atom are fermions, but we only give a brief discussion of the theory of atomic structure. The properties of atoms and molecules are not very similar to the properties of other many-fermion systems, since the nuclei, so much heavier than the electrons, affect their structure so profoundly. Atomic nuclei are also composed of fermions — protons and neutrons — and some of the ideas used in nuclear theory are borrowed from atomic theory. It is also possible to regard certain nuclei as composed of alpha-particles, which are bosons, but this is awkward for a number of reasons, and the alpha-particle model has only a limited range of usefulness. The valence electrons in a metal are also fermions, and it might be possible to produce plasmas whose properties are influenced strongly by the Pauli exclusion principle. The so-called "electron gas" serves as a model for either of these systems. A neutron star, a sphere of neutrons bound together by gravitational forces, is another many-fermion system; pulsars are widely believed to be neutron stars.
There is no example other than He4 of a system of many bosons of finite mass which displays quantum properties, but there are many systems that contain zero-mass bosons at finite temperatures. Phonons can exist in almost all systems, whatever sort of molecules they are composed of. Spin-waves exist in ferromagnetic or antiferromagnetic materials. In many-fermion systems at very low temperatures, there are phonons of "zero sound."
At first sight it would seem presumptuous to make any attempt to construct a detailed theory of such systems. In the first place, the quantum mechanics of even a three-body system is poorly understood, and long calculations are necessary to get any results for this apparently simpler problem. In the second place, although the qualitative properties of a classical gas or solid are well understood, the theory of liquids is very difficult and their properties are not fully understood. The systems we have mentioned are certainly not like solids, and do not have sufficiently low densities to be like gases, so it might be thought that the theory of such systems should be even more difficult than the theory of classical liquids, and that detailed and laborious calculations should yield only fragmentary and unconvincing results. This is in part true, but there is a brighter side to the picture, and it is to this side that we will pay most attention.
It has been known for a long time that it is a good approximation to regard the electrons in an atom as independent particles moving in a potential determined by the average positions of all the other electrons as well as by the nucleus. There are two reasons for the usefulness of this approximation. One is that the Coulomb potential has a long range, so that the position of a single electron does not greatly affect the potential at a point; the potential is determined more by the state of the whole atom than by the position of any particular electron. The other reason is that the correlations between electrons produced by the exclusion principle are more important than the correlations due to the mutual repulsion of the electrons. For the same reasons, the valence electrons in a metal can be treated as identical but otherwise independent particles moving in a potential due to the ions and to the average effects of the other electrons. In both cases, this is only a first approximation, and the correlations between the electrons must be taken into account to explain some of the detailed properties of the systems. This approximation, in which the wave function of the system is approximated by an antisymmetrized product of one-particle wave functions, is known as the independent particle model.
It was not expected from a theoretical point of view that nucleons in a nucleus should behave in this way, since the forces between nucleons are strong and have a range comparable with the distance between nearest neighbors in a nucleus. It was expected that a nucleus should be more similar to an ordinary liquid drop than to an atom. It was therefore quite surprising when the success of the shell model and the optical model showed that an independent particle model of nuclei could explain many facts which the liquid drop model could not explain. The situation became even more puzzling when it was found that the potential between nucleons, like the potential between molecules, becomes very strongly repulsive at short distances. Much of the recent work on many-body problems has been directed towards an understanding of why the independent particle model works so well, and what the relation is between the interaction used in a shell model calculation and the interaction between free nucleons observed in a nucleon-nucleon scattering experiment.
It would be too sanguine to say that we know the answer to this problem, but at least it is true that the situation is less puzzling than it was. It is clear that it is the exclusion principle that is largely responsible for the validity of the independent particle model, since it prevents many of the nucleon-nucleon scattering processes which could occur in the absence of other identical particles. It is also clear that the independent "particle" of the model need not be a real nucleon, but may be a nucleon surrounded by some distortion of the neighboring nuclear matter, and this entity is known as a "quasiparticle." This idea is one that is borrowed from field theory, since there the physical particles are by no means identical with the bare particles to which the field operators refer, but each physical particle consists of a bare particle together with its associated vacuum polarization. The physical vacuum in the theory of elementary particles can be regarded as a particular example of a many-body system, because of the virtual particles which exist in it. The bare particles correspond to the physical particles of the many-body problem, and the physical particles of field theory correspond to the quasiparticles of the many-body problem. A simpler illustration of this idea occurs in the theory of ionic solutions, where an ion whose mobility is measured may consist of the simple ion surrounded by a variable number of loosely attached water molecules. We might refer to the ion and its attached molecules as a quasiparticle. This idea has been worked out in some detail for the theory of liquid He3, and was later applied to the theory of nuclei.
The questions of how one should justify the independent particle model for a particular system, under what conditions the model may give a good description of a system, and of how one can make corrections to the model are not easy to answer. Most work on these questions has used perturbation theory, but this is unsatisfactory for a number of reasons, as we shall see in detail later. The main difficulty is that the higher-order terms in the perturbation series have such a complicated structure that only the first two or three terms in the series can be calculated. Any conjectures on whether the series converges or not, or on how fast it converges, must be very uncertain when based on so little evidence. This uncertainty is increased by the fact that, in the physical problems we consider, the ratio of those successive terms that we can calculate is usually at best 10:1, rather than the 100:1 which occurs in electrodynamics.
Even if the second term of the series is much larger than the first, it does not necessarily mean that the initial approximation is bad, but may mean only that the wrong parameter has been chosen for the expansion. For example, if there is a strong repulsive force at very short distances, an expansion in powers of the potential has very large terms in it, whereas the leading terms of a series in powers of the scattering length may be quite small. Much of the work which we discuss is concerned with this sort of question. The great advantage of perturbation theory is that it provides a systematic method of calculation, but one is never sure what is the meaning of results obtained in this way.
The use of dispersion relations in this field has been limited, but it has led to some interesting results. It is possible, by the use of dispersion relations, to say that from some of the assumptions of the independent particle model certain other, not obviously connected, consequences follow. It is also possible to make deductions about the nature of the nucleon-nucleon potential from experimental results on large nuclei. Dispersion relations cannot necessarily answer a particular interesting question, but where they can be applied they may lead to rigorous answers.
Another line of approach is to use our knowledge of classical physics and macroscopic physics to correct some of the obvious shortcomings of the independent particle model. The independent particle model ignores some of the conservation laws which we know must hold, and certain simple "collective" excited states look quite complicated when described in terms of the independent particle model. Keeping one eye on the independent particle model and the other on the results which we know to be true it is not difficult to correct these shortcomings. Some interesting modifications of the expected results have been derived in this way. Such qualitative results are the most important successes of many-body physics, rather than the detailed quantitative calculations.
These ideas are developed in the second, third, fourth, and fifth chapters of this book, and the methods are applied mainly to the problems of nuclear structure, liquid He3 at zero temperature, and, to a lesser extent, atomic structure. The sixth chapter is devoted to the theory of noninteracting fermions in an inhomogeneous medium. The methods used for such problems are similar to the methods used for interacting fermions in an infinite homogeneous system, and the results are sometimes, but not always, simpler for the former. Such problems arise in the theory of impurities in metals, and in the theory of alloys. In the following three chapters we show that the same ideas can be used, with slight modifications, also for calculating the properties of systems in thermal equilibrium at finite temperatures. The expressions obtained are slightly more complicated, because of the extra variable (temperature), and calculations are considerably more difficult, but there is no difference in principle. We apply these methods to the theory of the electron gas. The only new idea which is introduced here is that a quasiparticle may be a combination of a "particle" and a "hole," which is the basis of the modern theory of superconductivity. This idea is also of importance in the theory of nuclear structure, and may be applicable to He3. Otherwise, these three chapters are meant to illuminate more fully the ideas introduced in the earlier chapters.
The last chapter gives a brief discussion of the properties of many-boson systems. Most of the methods used for many-fermion systems can be applied to the many-boson problem, with slight modifications. The physical significance of the results is rather different, although the formalism is so similar. The very reason for the existence of quasiparticles, which is the basis of the similarity between many-fermion and many-boson systems, is different in the two cases. In a system such as liquid He4, a finite fraction of the particles condense in the zero-momentum state. These scatter against one another, forming pairs of particles with opposite momentum. A quasiparticle can be created either by putting a particle in a state with a particular momentum, or by removing a particle with the opposite momentum. The quasiparticle is therefore a combination of a particle and a hole, as in the theory of superconductivity. It is the particles condensed in the zero-momentum state which stabilize the quasiparticles and so make this approximation useful. It is somewhat misleading to talk about the symmetry of the wave function dominating the properties of such systems, since the wave function which gives the lowest energy must be symmetric. It is rather the wave nature of the particles which has such a strong influence on the properties of He. The properties of the low excited states are dominated by the symmetry requirement, since many low-lying states are forbidden by the symmetry. There is also a brief discussion of systems of interacting phonons, but no topic is treated in detail in this chapter.
In recent years numerous books have been published which are mainly or entirely devoted to the quantum theory of many-body systems. Discussions of basic techniques are given in the books by Schultz (1964), by Ziman (1969), and by Mattuck (1967). Comprehensive treatments have been given by Nozieres (1964), by Abrikosov et al. (1963), and by March et al. (1967). Problems in nuclear physics are discussed by Brown (1967) and by Kumar (1962), while applications to solid-state physics are discussed in the books by Pines (1963) and by Anderson (1963). The theory of quantum liquids has been treated in books by Pines and Nozieres (1966) and by Feenberg (1969).
(Continues…)
Excerpted from "The Quantum Mechanics of Many-Body Systems"
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Copyright © 1990 D. J. Thouless.
Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Preface to the Second Edition, ix,
Preface to the First Edition, xi,
I. INTRODUCTION, 1,
II. SOLUBLE MODELS,
1. Introduction, 7,
2. Noninteracting fermions and bosons, 8,
3. Second quantization, 9,
4. Density matrices, 13,
5. Harmonic forces, 16,
6. SU3, 18,
7. One-dimensional problems, 23,
III. VARIATIONAL METHODS,
1. The Hartree-Fock equations, 26,
2. The self-consistent field for atoms, 28,
3. The Thomas-Fermi method, 30,
4. Nuclear matter, 32,
5. The Hartree-Fock equations for extended systems, 35,
6. Neutron matter, 37,
7. Alternative solutions of the Hartree-Fock equations, 38,
8. Magnetism and exchange, 44,
9. Jastrow's method, 49,
10. The shell model, 51,
IV. PERTURBATION THEORY,
1. General discussion, 55,
2. The Goldstone-Hugenholtz graphical method, 58,
3. Wick's theorem, 61,
4. Linked graphs, 66,
5. Rules for calculating with graphs, 68,
6. Hartree-Fock energies, 72,
7. Brueckner theory, 73,
8. Brueckner theory for finite nuclei, 79,
9. Divergence of the A-matrix, 81,
V. LOW-LYING EXCITED STATES,
1. Green functions and collective variables, 83,
2. One-particle Green functions, 84,
3. Perturbation calculation of Green functions, 89,
4. The optical model, 93,
5. The Fermi liquid, 96,
6. Sound and zero sound, 100,
7. Collective motion, 103,
8. Generator coordinates and projection, 107,
9. Two-particle Green functions, 109,
10. Time-dependent Hartree-Fock theory, 113,
11. Application to the shell model, 118,
VI. IMPURITIES AND RANDOM SYSTEMS,
1. Introduction, 122,
2. Isolated impurities, 123,
3. Dynamical problems, 130,
4. Random impurities, 134,
5. Electrical conductivity and the Kubo formula, 137,
VII. STATISTICAL MECHANICS AND SUPERCONDUCTIVITY THEORY,
1. The partition function, 142,
2. Free fermions and bosons, 144,
3. Superconductivity, 146,
4. Model of the superconducting state, 149,
5. Superconducting state of a real metal, 153,
6. The variational principle for the partition function, 156,
7. Quasiparticle method, 158,
8. Superfluidity of liquid helium three, 166,
9. The effect of pairing on nuclear properties, 167,
VIII. PERTURBATION THEORY AT FINITE TEMPERATURES,
1. Perturbation theory in classical statistical mechanics, 170,
2. The Bloch equation, 172,
3. Linked graph expansion, 173,
4. Comparison with ground state perturbation theory, 176,
5. Expectation value of an operator, 178,
6. Classical limit of perturbation theory, 179,
IX. GREEN FUNCTIONS AT FINITE TEMPERATURES,
1. Excited states at finite temperatures, 181,
2. Calculation of Green functions by perturbation theory, 183,
3. Plasma oscillations, 185,
4. Correlation energy, 189,
5. Screening, 196,
6. Survey of alternative techniques, 198,
7. Collective modes in superconductors, 201,
8. Enhanced paramagnetism in metals and liquid He3, 203,
X. BOSONS,
1. Introduction, 208,
2. Liquid helium, 209,
3. Phonons, 217,
References, 221,
Index, 235,