Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Civil War Book Review

An Adiabatic Study of the Positronium Negative Ion and Other Three-Particle Systems.

Javier Botero

Published Jan 1, 1986• 0 citations
Publisher

Abstract

In an effort to unravel the dependence of the dynamics of three-bound-particle systems on the relative masses of the particles, an adiabatic treatment in hyperspherical coordinates is used to study the ground state and some doubly excited states of Ps('-) (positronium negative ion) and other three-particle systems, including H('-) and M('-) (muonium negative ion). Two different methods for the solution of the adiabatic eigenvalue equation are presented; one uses a "prediagonalized" basis of hyperspherical harmonics, and the other uses an asymptotic basis which consists of hydrogenic functions for the positronium. Potential curves for Ps('-) and M('-) in the ('1)S('e), ('1)P('o), ('3)P('e), ('3)P('o) and ('1)D('e) states and for H('-) in the ('1)P('o) states are presented. A method for calculating the probability density as a function of the hyperspherical coordinates (R,(alpha),(theta)(,12)) is introduced and used for the P states of Ps('-). This helps to understand the different symmetries and to distinguish among them. A remarkable linear dependence of the binding energy of the ground state and of the doubly excited states on the reduced mass of the atom, discovered in this study, is presented and discussed.

DOI
10.31390/gradschool_disstheses.4221

Related papers

Asymptotic Adiabatic Hyperspherical Potential Curves for Three Body Systems.

Zheng Zhen

The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.

•Jan 1, 1988•0 citations

Resonant Photodetachment of the Positronium Negative Ion

J. Botero, Chris H. Greene

An adiabatic treatment in hyperspherical coordinates is used to predict the existence of resonant structures in ${\mathrm{Ps}}^{\ensuremath{-}}$ (Ps denotes positronium) photodetachment. This highly correlated system displays a weak series of narrow Feshbach resonances lying just below the $\mathrm{Ps}(n=2)$ threshold, while a single strong shape resonance dominates the photodetachment spectrum above this threshold as in ${\mathrm{H}}^{\ensuremath{-}}$. Use of a "prediagonalized" representation of hyperspherical harmonics dramatically improves the convergence of the adiabatic calculation.

Physical Review Letters•Mar 31, 1986•72 citations

Bound-state properties of the positronium negative ion Ps−

Alexei M. Frolov

Various geometrical and energetical (bound-state) properties of the positronium negative ion $({\mathrm{Ps}}^{\ensuremath{-}}$ or ${e}^{\ensuremath{-}}{e}^{+}{e}^{\ensuremath{-}})$ are determined by using highly accurate variational wave functions. These wave functions have been constructed by applying the advanced two-stage strategy proposed by Frolov [Phys. Rev. A 57, 2436 (1998)]. The determined total energy of the ground state $E=\ensuremath{-}0.2620050702329757\mathrm{a}.\mathrm{u}.$ is the lowest and most accurate value obtained for this system to date (the corresponding binding energy equals $\ensuremath{-}0.326674721317821\mathrm{eV}).$ The computation of the second-order relativistic corrections $(\ensuremath{\simeq}{\ensuremath{\alpha}}^{2})$ to the total energy is discussed also. The general form of the Breit-Pauli Hamiltonian in the relative coordinates for the ${\mathrm{Ps}}^{\ensuremath{-}}$ ion is presented.

Physical Review A•Oct 1, 1999•72 citations

Three-body Coulomb bound states

A. K. Bhatia, Richard J. Drachman

We reexamine the binding energies of three-particle systems containing two electrons and one positive particle of mass M, in an attempt to understand the approximate proportionality of the $^{1}\mathrm{S}^{\mathrm{e}}$ ground-state binding energies to the reduced masses, as pointed out by Botero and Greene [Phys. Rev. 56, 1366 (1986)]. Our investigation involves a careful evaluation of the contribution to the energy of the mass-polarization term ${H}_{\mathrm{mp}=\mathrm{\ensuremath{-}}(2/\mathrm{M}){\ensuremath{\nabla}}_{1}\mathrm{\ensuremath{\cdot}}{\ensuremath{\nabla}}_{2}}$ which is the only part of the Hamiltonian not scaling like the reduced mass. It does not appear that any fundamental principle is involved, since we find merely that the mass polarization decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited $^{3}\mathrm{P}^{\mathrm{e}}$ state, this reduction is not sufficient to allow binding when M approaches the electron mass. As a by-product, we compute accurately some properties of the recently observed negative muonium ion (${e}^{\mathrm{\ensuremath{-}}}$${\ensuremath{\mu}}^{+}$${e}^{\mathrm{\ensuremath{-}}}$).

Physical Review A•May 1, 1987•44 citations

Low-energy scattering properties of ground-state and excited-state positronium collisions

Michael D. Higgins, K. M. Daily, Chris H. Greene

Low-energy elastic and inelastic scattering in the Ps($1s$)-Ps($2s$) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum $L=0$ and parity equal to $+1$. The $s$-wave scattering lengths for the asymptotic Ps($1s$)-Ps($2s$) channel are calculated for each spin configuration. Results obtained for the $s$-wave scattering lengths are ${a}_{\mathrm{TT}}=7.3(2){a}_{0}\ensuremath{-}i0.02(1){a}_{0}, {a}_{\mathrm{SS}}=13.2(2){a}_{0}\ensuremath{-}i0.9(2){a}_{0}$, and ${a}_{\mathrm{ST}}=9.7(2){a}_{0}$ for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross sections are carried out for Ps atoms initially prepared in different uncoupled spin states.

arXiv:1904.04295•Journal of Physics Conference Series•Jul 29, 2019•4 citations

Ultraslow p¯ -H collisions in hyperspherical coordinates: Hydrogen and protonium channels

Brett D. Esry, H. R. Sadeghpour

The collision of an antiproton and a hydrogen atom at ultralow energies is described in hyperspherical coordinates. An adiabatic approximation leads to potential-energy curves that can be used to describe the dynamics of capture in the $\overline{p}\ensuremath{-}\mathrm{H}(1s)$ collision. A unified formulation of the collision and capture processes leading to formation of $p\overline{p}$ bound states emerges. Both $\mathrm{Pn}{(\mathrm{nl})\ensuremath{-}e}^{\ensuremath{-}}$ ``atomic-type'' and $\overline{p}\ensuremath{-}\mathrm{H}{(n}^{\ensuremath{'}}{l}^{\ensuremath{'}})$ ``molecular-type'' states are represented. These potential curves are calculated to more than 10 000 protonium Bohr radii, where low-energy antiprotons are stopped in collisions with hydrogen atoms. Elastic and inelastic partial and total protonium formation cross sections in the mass-scaled $\overline{p}$-H system are calculated. The existence of protonium negative (or positive) ion in its ground and excited states is discussed.

Physical Review A•Jan 14, 2003•31 citations
In an effort to unravel the dependence of the dynamics of three-bound-particle systems on the relative masses of the particles, an adiabatic treatment in hyperspherical coordinates is used to study the ground state and some doubly excited states of Ps('-) (positronium negative ion) and other three-particle systems, including H('-) and M('-) (muonium negative ion). Two different methods for the solution of the adiabatic eigenvalue equation are presented; one uses a "prediagonalized" basis of hyperspherical harmonics, and the other uses an asymptotic basis which consists of hydrogenic functions for the positronium. Potential curves for Ps('-) and M('-) in the ('1)S('e), ('1)P('o), ('3)P('e), ('3)P('o) and ('1)D('e) states and for H('-) in the ('1)P('o) states are presented. A method for calculating the probability density as a function of the hyperspherical coordinates (R,(alpha),(theta)(,12)) is introduced and used for the P states of Ps('-). This helps to understand the different symmetries and to distinguish among them. A remarkable linear dependence of the binding energy of the ground state and of the doubly excited states on the reduced mass of the atom, discovered in this study, is presented and discussed.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.
An adiabatic treatment in hyperspherical coordinates is used to predict the existence of resonant structures in ${\mathrm{Ps}}^{\ensuremath{-}}$ (Ps denotes positronium) photodetachment. This highly correlated system displays a weak series of narrow Feshbach resonances lying just below the Ps(n=2)\mathrm{Ps}(n=2)Ps(n=2) threshold, while a single strong shape resonance dominates the photodetachment spectrum above this threshold as in ${\mathrm{H}}^{\ensuremath{-}}$. Use of a "prediagonalized" representation of hyperspherical harmonics dramatically improves the convergence of the adiabatic calculation.
Various geometrical and energetical (bound-state) properties of the positronium negative ion $({\mathrm{Ps}}^{\ensuremath{-}}$ or ${e}^{\ensuremath{-}}{e}^{+}{e}^{\ensuremath{-}})$ are determined by using highly accurate variational wave functions. These wave functions have been constructed by applying the advanced two-stage strategy proposed by Frolov [Phys. Rev. A 57, 2436 (1998)]. The determined total energy of the ground state $E=\ensuremath{-}0.2620050702329757\mathrm{a}.\mathrm{u}.$ is the lowest and most accurate value obtained for this system to date (the corresponding binding energy equals $\ensuremath{-}0.326674721317821\mathrm{eV}).$ The computation of the second-order relativistic corrections $(\ensuremath{\simeq}{\ensuremath{\alpha}}^{2})$ to the total energy is discussed also. The general form of the Breit-Pauli Hamiltonian in the relative coordinates for the ${\mathrm{Ps}}^{\ensuremath{-}}$ ion is presented.
We reexamine the binding energies of three-particle systems containing two electrons and one positive particle of mass M, in an attempt to understand the approximate proportionality of the 1Se^{1}\mathrm{S}^{\mathrm{e}}1Se ground-state binding energies to the reduced masses, as pointed out by Botero and Greene [Phys. Rev. 56, 1366 (1986)]. Our investigation involves a careful evaluation of the contribution to the energy of the mass-polarization term ${H}_{\mathrm{mp}=\mathrm{\ensuremath{-}}(2/\mathrm{M}){\ensuremath{\nabla}}_{1}\mathrm{\ensuremath{\cdot}}{\ensuremath{\nabla}}_{2}}$ which is the only part of the Hamiltonian not scaling like the reduced mass. It does not appear that any fundamental principle is involved, since we find merely that the mass polarization decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited 3Pe^{3}\mathrm{P}^{\mathrm{e}}3Pe state, this reduction is not sufficient to allow binding when M approaches the electron mass. As a by-product, we compute accurately some properties of the recently observed negative muonium ion (${e}^{\mathrm{\ensuremath{-}}}$${\ensuremath{\mu}}^{+}$${e}^{\mathrm{\ensuremath{-}}}$).
Low-energy elastic and inelastic scattering in the Ps(1s1s1s)-Ps(2s2s2s) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum L=0L=0L=0 and parity equal to +1+1+1. The sss-wave scattering lengths for the asymptotic Ps(1s1s1s)-Ps(2s2s2s) channel are calculated for each spin configuration. Results obtained for the sss-wave scattering lengths are ${a}_{\mathrm{TT}}=7.3(2){a}_{0}\ensuremath{-}i0.02(1){a}_{0}, {a}_{\mathrm{SS}}=13.2(2){a}_{0}\ensuremath{-}i0.9(2){a}_{0}$, and aST=9.7(2)a0{a}_{\mathrm{ST}}=9.7(2){a}_{0}aST​=9.7(2)a0​ for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross sections are carried out for Ps atoms initially prepared in different uncoupled spin states.
The collision of an antiproton and a hydrogen atom at ultralow energies is described in hyperspherical coordinates. An adiabatic approximation leads to potential-energy curves that can be used to describe the dynamics of capture in the $\overline{p}\ensuremath{-}\mathrm{H}(1s)$ collision. A unified formulation of the collision and capture processes leading to formation of pp‾p\overline{p}pp​ bound states emerges. Both $\mathrm{Pn}{(\mathrm{nl})\ensuremath{-}e}^{\ensuremath{-}}$ ``atomic-type'' and $\overline{p}\ensuremath{-}\mathrm{H}{(n}^{\ensuremath{'}}{l}^{\ensuremath{'}})$ ``molecular-type'' states are represented. These potential curves are calculated to more than 10 000 protonium Bohr radii, where low-energy antiprotons are stopped in collisions with hydrogen atoms. Elastic and inelastic partial and total protonium formation cross sections in the mass-scaled p‾\overline{p}p​-H system are calculated. The existence of protonium negative (or positive) ion in its ground and excited states is discussed.