As we have seen in Section 8.4, the basis of our perception of an objective reality is statistical mechanics and field theory. These are routinely used in equilibrium and nonequilibrium statistical thermodynamics (see, for example, Calzetta & Hu [59]), which govern the macroscopic laws of everyday life. Thus the quantum theory of fields must be the fundamental description of Nature. Therefore, the simpler quantum mechanics of particles is necessarily a derived description. This chapter discusses the extent to which a particle picture of matter and radiation is appropriate. In relativistic quantum field theory, particles appear only as asymptotic excitations of the fields. The quantum particle concept therefore makes mathematical (and intuitive) sense only under very special circumstances, namely in those where a system actually behaves like particles do: - As quasiparticles, they are useful in the domain, where the geometric optics perspective applies. - Before and after a scattering event in a dilute gas or a particle collider ring, particles can be considered as being essentially free. During the very short interaction time itself, the scattering process cannot be described by a particle picture, but needs a hydromechanical description or a kinetic description via Kadanoff-Baym equations. Already at the very beginning of modern quantum mechanics, when Born found his probability interpretation, the scattering process of single particles in an external classical field was understood (see Section 14.1 of the Appendix) to be a wave process relating initially and finally stationary particles, freely moving in their center of mass frame. It is a historical accident that one continues to use the name particle in the many microscopic situations, where it is grossly inappropriate to think of it with the classical meaning of a tiny bullet moving through space. When we restrict the use of the particle concept to where it is appropriate, or when we do not think of particles as “objects”, in both cases all mystery is gone, and the foundations become fully rational. The inappropriate focus on the particle aspect of quantum mechanics created the appearance of mystery; common sense is restored by focussing instead on the field aspect. To substantiate the above summary view, we give in Section 12.1 an analysis of the photoelectric effect and in Section 12.2 a review of the origin of particle tracks. This leads in Section 12.3 to a discussion about the reality of particles. In the remainder of the chapter, we discuss the relation between fields and particles in more detail. Section 12.4 poses the problem how the nonrelativistic multiparticle description arises from relativistic quantum field theory. Section 12.5 considers the case of free quantum fields, and Section 12.6 the case of interacting quantum fields. Section 12.7, the final section, discusses the semiclassical view of quasiparticles in the approximation familiar from geometric optics.
As we have seen in Section 8.4, the basis of our perception of an objective reality is statistical mechanics and field theory. These are routinely used in equilibrium and nonequilibrium statistical thermodynamics (see, for example, Calzetta & Hu [59]), which govern the macroscopic laws of everyday life. Thus the quantum theory of fields must be the fundamental description of Nature. Therefore, the simpler quantum mechanics of particles is necessarily a derived description. This chapter discusses the extent to which a particle picture of matter and radiation is appropriate. In relativistic quantum field theory, particles appear only as asymptotic excitations of the fields. The quantum particle concept therefore makes mathematical (and intuitive) sense only under very special circumstances, namely in those where a system actually behaves like particles do: - As quasiparticles, they are useful in the domain, where the geometric optics perspective applies. - Before and after a scattering event in a dilute gas or a particle collider ring, particles can be considered as being essentially free. During the very short interaction time itself, the scattering process cannot be described by a particle picture, but needs a hydromechanical description or a kinetic description via Kadanoff-Baym equations. Already at the very beginning of modern quantum mechanics, when Born found his probability interpretation, the scattering process of single particles in an external classical field was understood (see Section 14.1 of the Appendix) to be a wave process relating initially and finally stationary particles, freely moving in their center of mass frame. It is a historical accident that one continues to use the name particle in the many microscopic situations, where it is grossly inappropriate to think of it with the classical meaning of a tiny bullet moving through space. When we restrict the use of the particle concept to where it is appropriate, or when we do not think of particles as “objects”, in both cases all mystery is gone, and the foundations become fully rational. The inappropriate focus on the particle aspect of quantum mechanics created the appearance of mystery; common sense is restored by focussing instead on the field aspect. To substantiate the above summary view, we give in Section 12.1 an analysis of the photoelectric effect and in Section 12.2 a review of the origin of particle tracks. This leads in Section 12.3 to a discussion about the reality of particles. In the remainder of the chapter, we discuss the relation between fields and particles in more detail. Section 12.4 poses the problem how the nonrelativistic multiparticle description arises from relativistic quantum field theory. Section 12.5 considers the case of free quantum fields, and Section 12.6 the case of interacting quantum fields. Section 12.7, the final section, discusses the semiclassical view of quasiparticles in the approximation familiar from geometric optics.
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.
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
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.
This is the third of three chapters that apply the concepts of Chapters 1, 2, and 3 to the three principal types of waves: mechanical, electromagnetic, and quantum. Although mechanical waves may be the most obvious in everyday life, and electromagnetic waves may be the most useful to our technological society, a case can be made that quantum waves are the most fundamental. As you'll see in this chapter, every bit of matter in the Universe behaves like a wave under certain circumstances, so it's hard to imagine anything more fundamental. If you're thinking that life in the macroscopic world moves along just fine without considering quantum effects, you should realize that the laws of classical physics are approximations, and all the quantum weirdness described in this chapter blends smoothly with classical physics when they overlap. Much of that weirdness comes from the dual wave and particle nature of matter and energy, and one of the goals of this chapter is to help you understand what's actually doing all this waving. This chapter begins with a comparison of the characteristics of waves and particles in Section 6.1, followed by a discussion of wave–particle duality in Section 6.2. You can read about the Schrödinger equation and probability wavefunctions in Sections 6.3 and 6.4, and quantum wave packets are discussed in Section 6.5. Wave and particle characteristics Before studying modern physics, most students think of particles and waves as belonging to fundamentally different categories of objects. That's understandable, because particles and waves have several very different characteristics, including the way in which they occupy space, the way they travel through openings, and the way they interact with other particles or waves. Here's a summary of some of the differentiating characteristics between particles and waves. Occupying space. Particles exist in a well-defined amount of space; if you could pause time at some instant during a tennis match, all of the spectators would agree on the position of the ball at that instant.
In Chapter 1, we treated the electromagnetic field as classical. Henceforth, we will want to treat it as a quantum field. In order to do so, we will first present some of the formalism of quantum field theory. This formalism is very useful in describing many-particle systems and processes in which the number of particles changes. Why this is important to a quantum description of nonlinear optics can be seen by considering a parametric amplifier of the type discussed in the last chapter. A pump field, consisting of many photons, amplifies idler and signal fields by means of a process in which a pump photon splits into two lower-energy photons, one at the idler frequency and one at the signal frequency. Therefore, what we would like to do in this chapter is to provide a discussion of some of the basics of quantum field theory that will be useful in the treatment of the quantization of the electromagnetic field. In particular, we will begin with a summary of quantum theory notation, and a discussion of many-particle Hilbert spaces. These provide the arena in which all of the action takes place. We will then move on to a treatment of the canonical quantization procedure for fields. This will allow us to develop a scattering theory for fields, which is ideally what we need. This relates the properties of a field entering a medium to those of the field leaving it, and this corresponds to what is done in an experiment.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].