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Identical particles

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
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Abstract

Abstract This chapter discusses the case of a many-body system of identical particles. It considers the consequencies of the symmetry of the Hamiltonian under particle permutations in this case. It discusses how to build many-body Boson and Fermion states. It then introduces the formalism of so-called second quantization, which is usefu, in the case of identical particles. It also expresses the many-body Hamiltonian in terms of creation and annihilation operators.

DOI
10.1093/acprof:oso/9780198749349.003.0025

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Abstract This chapter discusses the case of a many-body system of identical particles. It considers the consequencies of the symmetry of the Hamiltonian under particle permutations in this case. It discusses how to build many-body Boson and Fermion states. It then introduces the formalism of so-called second quantization, which is usefu, in the case of identical particles. It also expresses the many-body Hamiltonian in terms of creation and annihilation operators.
In this chapter we describe a technique to deal with identical particles that is called second quantization . Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields , fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level. Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.
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.
This chapter describes a very elegant way of accounting for the symmetry of the states and the operators of systems of many identical particles and illustrates its use in a few simple calculations. It presents operators in identical particle systems that remove particles from and add particles to the system. This formal similarity is the reason the creation and annihilation operator formalism is called second quantization; one-particle wave functions appear to have become operators which create and annihilate particles, while single particle expectation values appear to have become operators for physical quantities. The Hanbury-Brown and Twiss experiment provides a simple way of observing this tendency of bosons to clump together. Basically, the experiment measures the probability of observing two photons simultaneously at different points in a beam of incoherent light. The actual measuring apparatus uses a half silvered mirror, to split the beam into some identical beams; this avoids the problem of one detector casting a shadow on the other.
Abstract Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.
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.