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American Journal of Physics

Identical particle states and operators and the case of two-electron scattering

W. R. Ferrell, R. H. Ritchie, Trinidad L. Ferrell

Published Oct 1, 1984• 5 citations
Publisher

Abstract

A simplified notation is introduced for dealing with systems of indistinguishable fermions and bosons. The second-quantized form of system operators is obtained in a simpler manner than is offered ordinarily in textbooks. Details of the calculation of the nonrelativistic M≂ller cross section for scattering of two electrons are given as an example.

DOI
10.1119/1.13794

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A simplified notation is introduced for dealing with systems of indistinguishable fermions and bosons. The second-quantized form of system operators is obtained in a simpler manner than is offered ordinarily in textbooks. Details of the calculation of the nonrelativistic M≂ller cross section for scattering of two electrons are given as an example.
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.
Second quantization in nonrelativistic quantum mechanics is given in a simplified way using eigenstates of the number operator. The field creation operator is shown to be the operator which adds a particle with a delta-function wave function to the system, while the field annihilation operator removes a particle with a delta-function wave function from the system. The field creation operator is proved to be the Hermitian conjugate of the annihilation operator. The commutation (anticommutation) relations for boson (fermion) field operators are proved from the symmetry (antisymmetry) of the many-particle wave function. The second quantized form of one- and two-particle operators is obtained from their form in wave mechanics using the definition of the field creation and annihilation operators.
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.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains