Phonon is undoubtedly one of the most important concepts in the physics of materials. Phonon is the result when we quantize vibrational field. Whereas Schrodinger’s wave function method is the most popular and intuitive method in doing the first quantization, one usually uses Heisenberg’s operator method in the second quantization. We feel that there is a methodological and pedagogical discontinuity here. So in this paper we will use Schrodinger’s wave functional method to quantize the lattice vibration to produce phonons. Wave functional is difficult due to the nondenumerably infinite number of dimensions of its domain. In this paper we will approximate this infinity through the discrete nature of crystalline lattice so that phonon can be represented by wave function with many variables
Phonon is undoubtedly one of the most important concepts in the physics of materials. Phonon is the result when we quantize vibrational field. Whereas Schrodinger’s wave function method is the most popular and intuitive method in doing the first quantization, one usually uses Heisenberg’s operator method in the second quantization. We feel that there is a methodological and pedagogical discontinuity here. So in this paper we will use Schrodinger’s wave functional method to quantize the lattice vibration to produce phonons. Wave functional is difficult due to the nondenumerably infinite number of dimensions of its domain. In this paper we will approximate this infinity through the discrete nature of crystalline lattice so that phonon can be represented by wave function with many variables
The Hamiltonian most often used to describe a system of electrons interacting with lattice vibrations has the asymmetry that the electrons are represented by the creation and annihilation operators which arise from quantum field theory whereas the lattice vibrations are represented by the one-particle ladder operators of the one-dimensional harmonic-oscillator problem. Although the equal spacing of the harmonic-oscillator energy levels and the commutation relations satisfied by the oscillator ladder operators have led to an interpretation of this formalism in terms of the "creation and annihilation" of phonons, the phonons thus defined are not the quasiparticles which are obtained by a field-theoretical formulation of the lattice Hamiltonian in the harmonic approximation. A Hamiltonian is derived in this paper in which both the electrons and the lattice (in the harmonic approximation) are represented in field-theoretical formalism. A lattice creation operator in this formulation actually creates a vibrational mode (in a given state) from the physical vacuum, in contrast to a ladder operator which merely generates transitions between two states of an already existing vibrational mode. As a consequence, the number operator for the lattice quasiparticles commutes with the Hamiltonian, a property which is useful in certain applications. It is shown that even though our Hamiltonian is formally different from the ordinary one, it yields equivalent results for the state vectors of physical interest.
It was shown in the chapter 4 that fluctuations of quantum fields are particles. The phonons in the solid state physics are a classical example: they are quanta of the crystal lattice vibrations. However, electrons or other elementary particles seem to be logically more primary than any field. Formally, one may introduce a field that upon quantization gives rise to the particles which really exist in our World. The field that possess the desired properties can be used as a theoretical object justified by the comparison of its predictions with experiment. This is possible in principal, but it is not strightforward. In this chapter, we would like to describe another method of second quantisation in which the elementary particles are considered as primary objects and the quantum field as a result. This approach allows us not only to complete logically the second quantisation presented in chapter 6, but also to introduce fermionic fields in a very natural way.
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.
By the time a student of physics is ready to tackle quantum mechanics he has become familiar with the classical harmonic vibrator through many examples, and knows the crucial role it played in the development of Planck's ideas. It is natural then to concentrate on the mathematical aspects of the harmonic oscillator equation in quantum mechanics, the solution of Schrödinger's equation, normalization of the wave-function, calculation of mean values and of matrix elements, leaving the physics to look after itself. At a more advanced level the harmonic vibrator provides the entry into field theories, second quantization etc., and as a general rule it tends to be viewed more as a vehicle for instruction in more interesting matters than as a physical system having its own considerable interest and importance. Here we shall seek to redress the balance and study the vibrator as a thing in itself, without losing sight of the variety of physical problems to which the results can be applied. One especially important set of applications, however, will get little attention at this stage – the vibrations of compound bodies and of extended physical systems provide such wealth of interest as to justify a volume to themselves; and it is to such matters that the whole of the third part of this work will be devoted. As soon as one begins to contemplate the harmonic vibrator one becomes aware of the exceptional nature of its behaviour, in that it conforms more closely than any other system to the classical rules.
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.
Some fundamental concepts and mathematical tools of the quant um theory of many particle systems, which are indispensable to the study of solid state physics. are presented. The concepts of collective excitations and quasi-particles are stressed. Concerning the mathematical tools, Quantum Mechanics is presented in its general, representation independent formalism. The many-particle problem is approached in the occupation number representation, or second quantization. Finally, as an application the interaction between electrons and the vibrations of a crystal lattice, described in terms of elementary excitations of collective waves, i.e., phonons, is expressed and discussed within this framework.