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Physical Review Letters

Second-Quantization Representation for Systems of Atoms, Nuclei, and Electrons

Marvin D. Girardeau

Published Nov 22, 1971• 57 citations
Publisher

Abstract

A many-atom representation has been derived in which bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus-nucleus, electron-electron, and nucleus-electron Coulomb interactions, but also atom-atom, atom-nucleus, and atom-electron Coulomb and exchange interactions, including breakup and recombination terms.

DOI
10.1103/physrevlett.27.1416

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nucl-thCross-listedarXiv:2604.22537•Apr 27, 2026
A many-atom representation has been derived in which bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus-nucleus, electron-electron, and nucleus-electron Coulomb interactions, but also atom-atom, atom-nucleus, and atom-electron Coulomb and exchange interactions, including breakup and recombination terms.
Presented herein is a second-quantized formulation for a system of interacting nuclei and electrons expressed in terms of the creation and annihilation operators of various species of atoms and ions which can be formed by combining various numbers of the "elementary" particles. Processes such as ionization and recombination appear naturally. Explicit expressions for interatomic forces including exchange effects are generated from a "cluster expansion" reminiscent of the Ursell expansion.
A method of constructing ’’atomic second quantization’’ representations is described, based on the introduction of redundant modes (’’ideal atom variables’’) which are then given physical content by carrying out a suitable unitary transformation, the ’’generalized Tani transformation.’’ In such a representation bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus–nucleus, electron–electron, and nucleus–electron Coulomb interactions, but also atom–atom, atom–nucleus, and atom–electron Coulomb and exchange interactions, including breakup and recombination terms. All possible scattering and reaction channels are exhibited simultaneously in this transformed Hamiltonian. The method is applicable to any species or mixture of species of composite particles, but for simplicity the derivation is restricted here to the case of atomic hydrogen.
The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
A previously developed representation for many-atom problems is extended by incorporation of all effects of interatomic exchange of electrons and nuclei into a ``projected Hamiltonian.'' An expression for the projected Hamiltonian is derived which is exact on the two-atom subspace, and hence incorporates those exchange effects which are important at low densities. By the use of the center-of-mass separation of atomic wavefunctions, this expression for the projected Hamiltonian is transformed into a representation in terms of atomic field operators ψν(r) and ψν†(r), where r is the center-of-mass position of an atom and v stands for all of its internal quantum numbers.
A practical method is developed to deal with the second quantization of the many-body system containing the composite particles.In our treatment,the modes associated with composite particles are regarded approximately as independent ones compared with those of unbound particles.The field operators of the composite particles thus arise naturally in the second quantization Hamiltonian.To be emphasized,the second quantization Hamiltonian has the regular structures which correspond clearly to different physical processes.
This work describes a few-body dynamics method based on the Faddeev integral equations in momentum space for determining the total cross sections of fusion and breakup reactions with two- and three-body final channels in the continuum, employing a cluster representation of the colliding nuclei. Total cross sections were obtained for the reactions 3He(T,D)4He^3\text{He}(T,D)^4\text{He}3He(T,D)4He, 3He(T,np)4He^3\text{He}(T,np)^4\text{He}3He(T,np)4He, 3He(T,nD)3He^3\text{He}(T,nD)^3\text{He}3He(T,nD)3He, 3He(3He,2p)4He^3\text{He}(^3\text{He},2p)^4\text{He}3He(3He,2p)4He, 3He(3He,pD)3He^3\text{He}(^3\text{He},pD)^3\text{He}3He(3He,pD)3He, 7Li(3He,04He)6Li^7\text{Li}(^3\text{He},\phantom{0}^4\text{He})^6\text{Li}7Li(3He,04He)6Li, 7Li(3He,D4He)4He^7\text{Li}(^3\text{He},D^4\text{He})^4\text{He}7Li(3He,D4He)4He, and 7Li(3He,T3He)4He^7\text{Li}(^3\text{He},T^3\text{He})^4\text{He}7Li(3He,T3He)4He, in which both the projectile and the target nucleus were treated in a cluster representation. The work also implements a two-potential method to determine the Coulomb t−t-t−matrix and to account for Coulomb effects in short-range dynamics in momentum space. Calculations of the initial-state Coulomb interaction were performed; furthermore, an estimate was obtained for the magnitude of the off-shell effect of the Coulomb t−t-t−matrix, as well as the magnitude of the atomic electron anti-screening effect on the Coulomb interaction of the colliding nuclei. The calculated contribution of the cluster mechanism to the total cross section of the considered fusion reactions in the kinetic energy range T∈[1 keV,20 MeV]T\in[1~\text{keV},20~\text{MeV}]T∈[1 keV,20 MeV] is in good agreement with known experimental data.