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Physical Review A

Reply to “Possibility of distinguishing between identical particles in quantum collision processes”

Igor Bray

Published Apr 1, 1999• 16 citations
Publisher

Abstract

Bencze and Chandler, in the preceding paper, are highly critical of the convergent close-coupling method as applied to the calculation of differential ionization processes. We argue that the criticism is unwarranted and arises from a misunderstanding of the close-coupling theory.
DOI
10.1103/physreva.59.3133

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Impossibility of distinguishing between identical particles in quantum collision processes

G. Bencze, Colston Chandler

The very successful convergent close-coupling (CCC) method [I. Bray and D. V. Fursa, Phys. Rev. A $54,$ 2991 (1996)] for calculating cross sections for the ionization of atoms by electron impact is examined for the case of hydrogen ionization. The theory is recast in an operator theoretic form, allowing explicit formulas to be given for the solutions of the truncated CCC equations. The limit to which these solutions tend as ever more terms are included in the truncation is shown to be the exact transition operator for the system. This analytical result is not consistent with the numerical CCC results, which yield amplitudes that violate the symmetrization principle of quantum mechanics and which depend for success on treating the electrons as classically, rather than quantally, indistinguishable particles. This new result is also inconsistent with the ``step function hypothesis'' [I. Bray, Phys. Rev. Lett. $78,$ 4721 (1997)] that was introduced to restore the consistency of the CCC calculations with the symmetrization postulate. In addition, the CCC method does not include long-range effects of the Coulomb interaction that are expected to be important in certain kinematic configurations at low energies. Consequently, it is not justified yet to interpret the numerical successes of the CCC method as confirmation of a fundamental dynamical understanding of electron-induced atomic ionization.

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Hyperspherical close-coupling study of hyperfine transitions in low-energy p+pμ and e±+Ps scattering

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We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}+\mathrm{Ps}$ (or equivalently, ${e}^{+}+\mathrm{Ps})$ scattering at low collision energies. The spin-spin interactions ${V}_{S}$ are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy $\ensuremath{\Delta}\ensuremath{\epsilon}$ in the separated-atom limit. A comparison is made with a simpler method in which the effect of ${V}_{S}$ is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of ${V}_{S}$ is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$ for both $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}$ collisions. The elastic cross sections are less sensitive to ${V}_{S},$ but those for $p+p\ensuremath{\mu}$ are still affected by ${V}_{S}$ for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$.

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Reply to ‘‘Note on ‘Bell’s theorem and the foundation of quantum physics’ ’’ [Am. J. Phys. 5 6, 565 (1988)]

Henry Pierce Stapp

American Journal of Physics•Jun 1, 1988•4 citations
The very successful convergent close-coupling (CCC) method [I. Bray and D. V. Fursa, Phys. Rev. A 54,54,54, 2991 (1996)] for calculating cross sections for the ionization of atoms by electron impact is examined for the case of hydrogen ionization. The theory is recast in an operator theoretic form, allowing explicit formulas to be given for the solutions of the truncated CCC equations. The limit to which these solutions tend as ever more terms are included in the truncation is shown to be the exact transition operator for the system. This analytical result is not consistent with the numerical CCC results, which yield amplitudes that violate the symmetrization principle of quantum mechanics and which depend for success on treating the electrons as classically, rather than quantally, indistinguishable particles. This new result is also inconsistent with the ``step function hypothesis'' [I. Bray, Phys. Rev. Lett. 78,78,78, 4721 (1997)] that was introduced to restore the consistency of the CCC calculations with the symmetrization postulate. In addition, the CCC method does not include long-range effects of the Coulomb interaction that are expected to be important in certain kinematic configurations at low energies. Consequently, it is not justified yet to interpret the numerical successes of the CCC method as confirmation of a fundamental dynamical understanding of electron-induced atomic ionization.
We consider the nonrelativistic N-body scattering problem for a system of particles in which some subsets of the particles are identical. We demonstrate how the particle identity can be included in a general class of linear integral equations for scattering operators or components of scattering operators. The Yakubovskii, Yakubovskii–Narodestkii, Rosenberg, and Bencze–Redish–Sloan equations are included in this class. Algebraic methods are used which rely on the properties of the symmetry group of the system. Operators depending only on physically distinguishable labels are introduced and linear integral equations for them are derived. This procedure maximally reduces the number of coupled equations while retaining the connectivity properties of the original equations.
We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}+\mathrm{Ps}$ (or equivalently, e++Ps){e}^{+}+\mathrm{Ps})e++Ps) scattering at low collision energies. The spin-spin interactions VS{V}_{S}VS​ are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy $\ensuremath{\Delta}\ensuremath{\epsilon}$ in the separated-atom limit. A comparison is made with a simpler method in which the effect of VS{V}_{S}VS​ is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of VS{V}_{S}VS​ is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$ for both $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}$ collisions. The elastic cross sections are less sensitive to VS,{V}_{S},VS​, but those for $p+p\ensuremath{\mu}$ are still affected by VS{V}_{S}VS​ for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$.