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American Journal of Physics

Introducing Electron Creation and Annihilation Operators in Solid State Physics

M. I. Darby

Published Feb 1, 1972• 0 citations
Publisher

Abstract

A brief and simple introduction to the theory of many (N) electron systems that is suitable for undergraduates is given. By considering only three electrons (N = 3) it is possible to write out all the results explicitly. The manipulations of both determinantal wavefunctions and wave-functions in the number representation are considered, and the electron annihilation and creation operators are defined. It is then shown how the Hamiltonian for N electrons is written in second quantized form.

DOI
10.1119/1.1986509

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Chapter 2

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A brief and simple introduction to the theory of many (N) electron systems that is suitable for undergraduates is given. By considering only three electrons (N = 3) it is possible to write out all the results explicitly. The manipulations of both determinantal wavefunctions and wave-functions in the number representation are considered, and the electron annihilation and creation operators are defined. It is then shown how the Hamiltonian for N electrons is written in second quantized form.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.