The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focusses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti)ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body quasi-particle space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general. In fact, second quantization can be considered the first major cornerstone on which the theoretical framework of quantum field theory was built.
The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focusses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti)ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body quasi-particle space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general. In fact, second quantization can be considered the first major cornerstone on which the theoretical framework of quantum field theory was built.
The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focuses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti) ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general.
This chapter introduces and applies the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. Second quantization formalism treats systems of bosons or fermions with a fixed or variable number of particles in a unified way. The chapter reviews the key aspects of this formalism. The formalism of second quantization treats systems of bosons with a fixed or variable number of particles in a unified manner. In many physical processes, the particle number does change. Different quantum operators are expressed most naturally in different representations, which make basis changes an essential issue in quantum physics. Quantum field theory is a universal tool that has applications ranging from atomic, molecular, and particle physics to condensed matter and statistical physics as well as modern quantum chemistry. The fundamental difference between classical and quantum mechanics relates to the concept of indistinguishability of identical particles.
Nothing can be made out of nothing. William Shakespeare , King Lear Historically, quantization of the motion of particles was developed first. The state was described by a wave function and observables by operators. When dealing with interactions between particles and fields, such as the electromagnetic field, the fields were treated classically. Classical field equations look like the quantum mechanical equations for the wave function of the field quanta. For example, the Klein–Gordon classical field equation is similar to the quantum mechanical wave equation for a relativistic spinless particle. Quantizing the fields, leading to quantum field theory, appears to be quantizing a theory that has already been quantized; hence the name “second quantization.” In reality, there is only one quantization and one quantum theory. The method of second quantization is important in the study of many-particle systems. It enables us to express many-body operators in terms of creation and annihilation operators, thus rendering calculations less cumbersome. Moreover, the method makes it possible to treat systems with a variable number of particles; that is why the method initially emerged in the context of quantum field theory. In Chapter 1 we indicated that any one-particle wave function may be expanded in a complete set of states. In this chapter, we show that products of single-particle states, when properly symmetrized, form an orthonormal basis for the expansion of the wave function of an N -particle system.
Abstract Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.
Up to a point, the quantum theory developed in Chapter 5 was quite general. However, the systems we had in mind were non-relativistic ones consisting of a xed number of particles. In this chapter, we extend the theory to deal with systems in which the number of particles can change. There are several reasons for wanting to do this. The most obvious is that we need a method of describing high-energy scattering and decay processes in which particles can be created and destroyed. A second is that, when we try to make quantum theory consistent with special relativity, we encounter diculties (discussed in Chapter 7) that can be resolved only in this more general setting. The nal reason is that, even for systems of non-relativistic particles, the mathematics rapidly becomes intractable as the number of particles increases. A useful device for dealing with large systems is, roughly speaking, to imagine adding an extra particle, which serves as a theoretical probe of the state of the system. To put the matter another way, the method of second quantization developed in this chapter provides a means of dealing with the entire system by considering only a few particles at a time.
Up to a point, the quantum theory developed in chapter 5 was quite general. However, the systems we had in mind were non-relativistic ones consisting of a fixed number of particles. In this chapter, we extend the theory to deal with systems in which the number of particles can change. There are several reasons for wanting to do this. The most obvious is that we need a method of describing high-energy scattering and decay processes in which particles can be created and destroyed. A second is that, when we try to make quantum theory consistent with special relativity, we encounter difficulties (discussed in chapter 7) that can be resolved only in this more general setting. The final reason is that, even for systems of non-relativistic particles, the mathematics rapidly becomes intractable as the number of particles increases. A useful device for dealing with large systems is, roughly speaking, to imagine adding an extra particle, which serves as a theoretical probe of the state of the system. To put the matter another way, the method of second quantization developed in this chapter provides a means of dealing with the entire system by considering only a few particles at a time.