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Routledge eBooks

Second Quantization. Single Nucleon and Two-Body Operators

Igal Talmi

Published Oct 6, 2017• 0 citations
Publisher

Abstract

In Section 14 we considered many nucleon states as the antisymmetrized products of single nucleon wave functions. Due to the antisymmetrization, the wave function is completely determined by the single nucleon states which are occupied. This suggests that a direct way to deal with many nucleon states is by using a representation in which occupied single nucleon states, rather than nucleon coordinates, are specified. To do this we have to define a fixed set of single nucleon states. The results ( 14.6 ), ( 14.7 ) indicate that we should provide matrix elements of relevant single nucleon operators, calculated in the basis of these single nucleon wave functions. The results ( 14.11 ), ( 14.12 ) and ( 14.13 ) show that we should also calculate matrix elements of two-body operators in the basis of single nucleon wave functions. These calculations should be performed in the way shown in preceding sections. They are carried out in configuration space by using wave functions of space coordinates as well as spin (and isospin) variables. Equipped with these matrix elements we can introduce the representation in terms of occupied states by using the formalism of second quantization. This formalism will now be briefly described without entering into serious discussions and rigorous proofs.

DOI
10.1201/9780203739716-17

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In Section 14 we considered many nucleon states as the antisymmetrized products of single nucleon wave functions. Due to the antisymmetrization, the wave function is completely determined by the single nucleon states which are occupied. This suggests that a direct way to deal with many nucleon states is by using a representation in which occupied single nucleon states, rather than nucleon coordinates, are specified. To do this we have to define a fixed set of single nucleon states. The results ( 14.6 ), ( 14.7 ) indicate that we should provide matrix elements of relevant single nucleon operators, calculated in the basis of these single nucleon wave functions. The results ( 14.11 ), ( 14.12 ) and ( 14.13 ) show that we should also calculate matrix elements of two-body operators in the basis of single nucleon wave functions. These calculations should be performed in the way shown in preceding sections. They are carried out in configuration space by using wave functions of space coordinates as well as spin (and isospin) variables. Equipped with these matrix elements we can introduce the representation in terms of occupied states by using the formalism of second quantization. This formalism will now be briefly described without entering into serious discussions and rigorous proofs.
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of 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 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.)
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.
A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.
Abstract The need to enforce fermionic antisymmetry in the nuclear many-body problem commonly requires use of single-particle coordinates, defined relative to some fixed origin. To obtain physical operators which nonetheless act on the nuclear many-body system in a Galilean-invariant fashion, thereby avoiding spurious center-of-mass contributions to observables, it is necessary to express these operators with respect to the translational intrinsic frame. Several commonly-encountered operators in nuclear many-body calculations, including the magnetic dipole and electric quadrupole operators (in the impulse approximation) and generators of U(3) and Sp ( 3 , R ) symmetry groups, are bilinear in the coordinates and momenta of the nucleons and, when expressed in intrinsic form, become two-body operators. To work with such operators in a second-quantized many-body calculation, it is necessary to relate three distinct forms: the defining intrinsic-frame expression, an explicitly two-body expression in terms of two-particle relative coordinates, and a decomposition into one-body and separable two-body parts. We establish the relations between these forms, for general (non-scalar and non-isoscalar) operators bilinear in coordinates and momenta.