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Physical Review A

Coulomb three-body problem in the second quantized form

Alexei M. Frolov

Published Sep 11, 2001• 4 citations
Publisher

Abstract

The Hamiltonian for an arbitrary Coulomb three-body system is written in closed form in terms of nine generators of three independent o(2,1) algebras and its six physical parameters (i.e., three masses and three charges). Moreover, it is shown that such a Hamiltonian can also be written in the second quantized form by using six pairs of creation and annihilation (boson) operators. The obtained finite-term representation of the Coulomb three-body Hamiltonian in the second quantized form is exact, i.e., it is not based on any approximation. In particular, this means that the original Coulomb three-body problem can be reduced to the equivalent problem of three interacting, complex (i.e., charged) boson fields. The developed procedure opens new avenues in obtaining analytical and semianalytical (or highly accurate) solutions for various Coulomb three-body problems.

DOI
10.1103/physreva.64.042503

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The Hamiltonian for an arbitrary Coulomb three-body system is written in closed form in terms of nine generators of three independent o(2,1) algebras and its six physical parameters (i.e., three masses and three charges). Moreover, it is shown that such a Hamiltonian can also be written in the second quantized form by using six pairs of creation and annihilation (boson) operators. The obtained finite-term representation of the Coulomb three-body Hamiltonian in the second quantized form is exact, i.e., it is not based on any approximation. In particular, this means that the original Coulomb three-body problem can be reduced to the equivalent problem of three interacting, complex (i.e., charged) boson fields. The developed procedure opens new avenues in obtaining analytical and semianalytical (or highly accurate) solutions for various Coulomb three-body problems.
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 Hookean model of a three-body problem for particles with arbitrary masses and charges where two of them interact with each other through a Coulomb potential and with the third through a harmonic potential is presented. It is shown that a condition relating the masses to the harmonic coupling constants must be satisfied in order to render this problem separable. A general exact analytic solution written in terms of the relative interparticle coordinates is given as well as general expressions for the total and binding energies of this three-body system. We apply these results to examine electronic, muonic, antiprotonic, and pionic families of non-Born-Oppenheimer Hookean systems. The first contains the atoms or atomic ions: ${\mathrm{Ps}}^{\ensuremath{-}}({e}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{H}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{D}}^{\ensuremath{-}}({d}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{T}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, $^{4}\mathrm{He}(h{e}^{+2}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, and the following molecular ions: ${\mathrm{Ps}}_{2}^{+}({e}^{\ensuremath{-}}{e}^{+}{e}^{+})$, ${\mathrm{H}}_{2}^{+}({e}^{\ensuremath{-}}{p}^{+}{p}^{+})$, $\mathrm{H}{\mathrm{D}}^{+}({e}^{\ensuremath{-}}{d}^{+}{p}^{+})$, ${\mathrm{HT}}^{+}({e}^{\ensuremath{-}}{t}^{+}{p}^{+})$, ${\mathrm{DT}}^{+}({e}^{\ensuremath{-}}{d}^{+}{t}^{+})$, ${\mathrm{D}}_{2}^{+}({e}^{\ensuremath{-}}{d}^{+}{d}^{+})$, ${\mathrm{T}}_{2}^{+}({e}^{\ensuremath{-}}{t}^{+}{t}^{+})$. The muonic and antiprotonic families are similar to the electronic ones except that the species are formed replacing ${e}^{\ensuremath{-}}$ by ${\ensuremath{\mu}}^{\ensuremath{-}}$ or ${p}^{\ensuremath{-}}$. The pionic family comprises exotic atoms containing at least one pion. We also apply these results to two-electron three-dimensional spherical quantum dots and for these systems we examine the effect of electronic correlation, particularly on the singlet-triplet transitions and on the collective motion of the electrons and center of mass leading to ``floppy''dynamics.
The emphasis of this work is on the computation of physical properties as well as of the wave function of interacting bosons in a trap potential. Many-body perturbation theory is employed to study the leading term of these quantities for finite numbers of bosons, and exact solutions are aimed at in the infinite-particle limit. As discussed before, a suitable starting point is the second-quantized Hamiltonian represented in the basis of destruction and creation operators of its own mean-field potential. This choice leads to expressions for the perturbation terms of all quantities which exhibit a very weak dependence on the particle number. Importantly, when applying ideas similar to Bogoliubov's, the Hamiltonian can be reduced in the infinite-particle limit to a much simplified form which is a priori particle-number conserving. The resulting phonon Hamiltonian is diagonalizable by a linear transformation for which an explicit eigenvalue equation is given. Physical properties can be expressed explicitly by elements of this transformation, and of particular relevance is that the particle-number-conserving wave functions of the original many-boson system can be reconstructed using recursion relations. The reconstruction of the particle-conserving wave function from the phonon Hamiltonian can also be used to assess when the infinite-particle limit is reached in practice for finite trapped condensates. Two applications are discussed in detail. For one of them, an exact solution is known which is found, in the infinite-particle limit, to exactly coincide with that of the phonon Hamiltonian. In both examples expressions for the properties are given in closed form. The physics behind the phonon Hamiltonian and its physical properties is discussed.
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.
For the purpose of investigating the N-Body Bose System with a finite number of states, we analyze Xij = Ai†Aj, i, j ≤ K. (A and A† are the usual annihilation and creation operators of the second quantization formalism). The Hamiltonian for a fixed number of particles may be expressed in terms of the (finite-dimensional) irreducible representations of the Xij. A set of fundamental equations is defined for the irreducible representations Xij(N, K), i, j ≤ K and is solved for arbitrary N and K. The analysis of the structure of the Xij(N, K) yields a simple, systematic method for listing all possible ways in which n1+ … ni+ … + nK = N may be satisfied for ni positive integral or zero. This leads to a particularly simple method for constructing the Xij(N, K), i, j ≤ K.