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Physical Review A

Exact non-Born-Oppenheimer wave functions for three-particle Hookean systems with arbitrary masses

Xabier López, Jesús María Ugalde, Lorenzo Echevarría, Eduardo V. Ludeña

Published Oct 6, 2006• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.74.042504

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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: Ps\ensuremath−(e+e\ensuremath−e\ensuremath−){\mathrm{Ps}}^{\ensuremath{-}}({e}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})Ps\ensuremath−(e+e\ensuremath−e\ensuremath−), H\ensuremath−(p+e\ensuremath−e\ensuremath−){\mathrm{H}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})H\ensuremath−(p+e\ensuremath−e\ensuremath−), D\ensuremath−(d+e\ensuremath−e\ensuremath−){\mathrm{D}}^{\ensuremath{-}}({d}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})D\ensuremath−(d+e\ensuremath−e\ensuremath−), T\ensuremath−(p+e\ensuremath−e\ensuremath−){\mathrm{T}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})T\ensuremath−(p+e\ensuremath−e\ensuremath−), 4He(he+2e\ensuremath−e\ensuremath−)^{4}\mathrm{He}(h{e}^{+2}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})4He(he+2e\ensuremath−e\ensuremath−), and the following molecular ions: Ps2+(e\ensuremath−e+e+){\mathrm{Ps}}_{2}^{+}({e}^{\ensuremath{-}}{e}^{+}{e}^{+})Ps2+​(e\ensuremath−e+e+), H2+(e\ensuremath−p+p+){\mathrm{H}}_{2}^{+}({e}^{\ensuremath{-}}{p}^{+}{p}^{+})H2+​(e\ensuremath−p+p+), HD+(e\ensuremath−d+p+)\mathrm{H}{\mathrm{D}}^{+}({e}^{\ensuremath{-}}{d}^{+}{p}^{+})HD+(e\ensuremath−d+p+), HT+(e\ensuremath−t+p+){\mathrm{HT}}^{+}({e}^{\ensuremath{-}}{t}^{+}{p}^{+})HT+(e\ensuremath−t+p+), DT+(e\ensuremath−d+t+){\mathrm{DT}}^{+}({e}^{\ensuremath{-}}{d}^{+}{t}^{+})DT+(e\ensuremath−d+t+), D2+(e\ensuremath−d+d+){\mathrm{D}}_{2}^{+}({e}^{\ensuremath{-}}{d}^{+}{d}^{+})D2+​(e\ensuremath−d+d+), T2+(e\ensuremath−t+t+){\mathrm{T}}_{2}^{+}({e}^{\ensuremath{-}}{t}^{+}{t}^{+})T2+​(e\ensuremath−t+t+). The muonic and antiprotonic families are similar to the electronic ones except that the species are formed replacing e\ensuremath−{e}^{\ensuremath{-}}e\ensuremath− by \ensuremathμ\ensuremath−{\ensuremath{\mu}}^{\ensuremath{-}}\ensuremathμ\ensuremath− or p\ensuremath−{p}^{\ensuremath{-}}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 wave functions of boson and fermion gases are known even when the particles have harmonic interactions. Here we generalize these results by solving exactly the N–body Schrödinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in three dimensions. For the harmonic case that function is linear in r2. Explicit N–body solutions are given when U(r) = –2Mh–2V(r) = 𝛇r–1 – 𝛇2r–2. Here M is the sum of the masses and r2 = 1/2 M–2ΣmImJ(xI – xJ)2. For general U(r) the solution is given in terms of the one– or two–body problem with potential U(r) in three dimensions. The degeneracies of the levels are derived for distinguishable particles, for bosons of spin zero and for spin 1/2 fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N–body wave function. The N–body forces involved in these extraordinary N–body problems are not the usual sums of two–body interactions, but nor are forces between quarks or molecules. Bose–Einstein condensation of particles in three dimensions interacting via these strange potentials can be treated by this method.
We consider three-body systems in two dimensions with zero-range interactions for general masses and interaction strengths. The momentum-space Schrödinger equation is solved numerically and in the Born-Oppenheimer (BO) approximation. The BO expression is derived using separable potentials and yields a concise adiabatic potential between the two heavy particles. The BO potential is Coulomb-like and exponentially decreasing at small and large distances, respectively. While we find similar qualitative features to previous studies, we find important quantitative differences. Our results demonstrate that mass-imbalanced systems that are accessible in the field of ultracold atomic gases can have a rich three-body bound state spectrum in two dimensional geometries. Small light-heavy mass ratios increase the number of bound states. For 87Rb-87Rb-6Li and 133Cs-133Cs-6Li we find respectively 3 and 4 bound states.
For pt.II see Proc. R. Soc. London, Ser. A, vol.384, no.1786, p.89-105 (1982). A procedure for solving the few-particle Schrodinger equation exactly is applied to a model system consisting of two identical particles and a massive third particle. The type of interaction potential is not specified except that it should not diverge more rapidly than r -2 at the particle positions. Allowable interactions include the Coulomb and the harmonic oscillator potentials. The principles are illustrated by reference to the spatially symmetric states of the system. The solution has the form of a multipole expansion in spherical polar coordinates. The radial dependence for each multipole component is defined by a power series including logarithmic terms. Explicit expressions are given for the coefficients in the expansion. Exact coefficients are obtained for low-order terms in the wavefunction of a two-electron atom.
The energy spectrum, atom-dimer scattering length, and atom-trimer scattering length for systems of three and four ultracold atoms with $\ensuremath{\delta}$-function interactions in one dimension are presented as a function of the relative mass ratio of the interacting atoms. The Born-Oppenheimer approach is used to treat three-body (``HHL'') systems of one light and two heavy atoms, as well as four-body (``HHHL'') systems of one light and three heavy atoms. Zero-range interactions of arbitrary strength are assumed between different atoms, but the heavy atoms are assumed to be noninteracting among themselves. Fermionic and bosonic heavy atoms with both positive and negative parity are considered.
For a three-electron system with finite-strength interactions confined to a one-dimensional harmonic trap, we solve the Schrödinger equation analytically to obtain the exact solutions, from which we construct explicitly the simultaneous eigenstates of the energy and total spin for the first time. The solutions for the three-electron system allow us to derive analytic expressions for the exact one-particle Green's function (GF) for the corresponding two-electron system. We calculate the GF in frequency domain to examine systematically its behavior depending on the electronic interactions. We also compare the pole structure of non-interacting GF using the exact Kohn-Sham (KS) potential with that of the exact GF to find that the discrepancy of the energy gap between the KS system and the original system is larger for a stronger interaction. We perform numerical examination on the behavior of GFs in real space to demonstrate that the exact and KS GFs can have shapes quite different from each other. Our simple model will help to understand generic characteristics of interacting GFs.
The quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center-of-mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl],with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi statistics, a = −½ for Bose statistics. The eigenfunctions corresponding to these energy levels are not obtained explicitly, but they are rather fully characterized. A more general model is similarly solved, in which the N particles are divided in families, with the same quadratic interaction acting between all pairs, but with the inversely quadratic interaction acting only between particles belonging to the same family, with a strength that may be different for different families. The second model, characterized by the pair potential g(xi − xj)−2, g > −ℏ2/(4m), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities xi ≥ xi.1, i = 1, 2, ⋯, N = 1, to which attention may be restricted without loss of generality) by (initial) momenta pi, i = 1, 2, ⋯, N, goes over into a final configuration characterized uniquely by the (final) momenta pi′, with pi′=pN+1−i. This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).