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American Journal of Physics

One-body and two-body operators on systems of identical particles

N. I. Greenberg, S. Raboy

Published Feb 1, 1982• 4 citations
Publisher

Abstract

The form of one-particle and two-particle operators is obtained in the notation of second quantization. These forms are applied to many-particle states of identical particles where the appropriate symmetry with respect to exchange of particles is invoked from the start. The form in the wave-mechanical formalism of operators suggested by classical analog is derived in a matter suitable for inclusion in courses in quantum mechanics.

DOI
10.1119/1.13033

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The form of one-particle and two-particle operators is obtained in the notation of second quantization. These forms are applied to many-particle states of identical particles where the appropriate symmetry with respect to exchange of particles is invoked from the start. The form in the wave-mechanical formalism of operators suggested by classical analog is derived in a matter suitable for inclusion in courses in quantum mechanics.
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.
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.
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.