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Physical Review A

Impossibility of distinguishing between identical particles in quantum collision processes

G. Bencze, Colston Chandler

Published Apr 1, 1999• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.59.3129

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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.
Quantum mechanics (QM) and quantum electrodynamics (QED) are often touted as the most successful theories ever. In this paper, this claim is critically evaluated by a test of internal consistency for the ability to calculate the conjugate observables of the nature of the free electron, ionization energy, elastic electron scattering, and the excited states of the helium atom using the same solution for each of the separate experimental measurements. It is found that in some cases quantum gives good numbers, but the solutions are meaningless numbers since each has no relationship to providing an accurate physical model. Rather, the goal is to mathematically reproduce an experimental or prior theoretical number using adjustable parameters including arbitrary wave functions in computer algorithms with precision that is often much greater (e.g. 8 significant figures greater) than possible based on the propagation of errors in the measured fundamental constants implicit in the physical problem. Given the constraints of adherence to physical laws and internal consistency, an extensive literature search indicates that quantum mechanics has never solved a single physical problem correctly including the hydrogen atom and the next member of the periodic chart, the helium atom. Rather than using postulated unverifiable theories that treat atomic particles as if they were not real, physical laws are now applied to the same problem. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of − e moving in the Coulombic field of the proton and the wave equation as modified after Schrodinger, a classical approach is explored which yields a model which is remarkably accurate and provides insight into physics on the atomic level. The proverbial view deeply seated in the wave-particle duality notion that there is no large-scale physical counterpart to the nature of the electron is shown not to be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantum atomic problems. Specifically, a theory of classical quantum mechanics (CQM) was derived from first principles as reported previously [1-7] that successfully applied physical laws to the solution of atomic problems that has its basis in a breakthrough in the understanding of the stability of the bound electron to radiation. Rather than using the postulated Schrodinger boundary condition: “ 0 Ψ→ as →∞ r ,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound 1 = n -state electron cannot radiate energy. Although it is well known that an accelerated point particle radiates, an extended distribution, modeled as a superposition of accelerating charges, does not have to radiate. A simple invariant physical model arises naturally wherein the predicted results are extremely straightforward and internally consistent requiring minimal math as in the case of the most famous equations of Newton, Maxwell, Einstein, de Broglie, and Planck on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used. The accurate solution of the helium atom is confirmed by the agreement of predicted and observed conjugate parameters using the same unique physical model in all cases.
The cross-section formula of Flannery [Phys. Rev. A 22, 2408 (1980)] in the semiquantal approximation for the processes referred to in the title is rewritten so as to make it more useful in practice. The formula is shown to be further simplified by taking an average over the azimuthal quantum number of a highly excited hydrogenic atom to be ionized. Numerical applications with use of the Glauber amplitude for the electron-atom inelastic scattering are made to the ionization collisions between two excited hydrogen atoms with simultaneous excitation and deexcitation of one of the atoms. The results are compared with those obtained by using the Born amplitude, and are analyzed in terms of the Glauber generalized oscillator strengths.
A general approximate method for computing the cross sections for certain inelastic collision processes is developed. It is based on the possibility of obtaining, with the use of the impact-parameter method, an exact expression for the transition amplitude in a special case. The dependence of the cross section on the parameters of coupling constant, velocity, and energy difference between states is discussed for a two-level model. Detailed calculations are made of cross section for collisions of the type $A(^{2}P_{\frac{3}{2}})+B\ensuremath{\rightleftharpoons}A(^{2}P_{\frac{1}{2}})+B$, where AAA is an excited atom of an alkali metal and BBB is an atom of a noble gas. An Appendix contains a review of the effective long-range interaction of atoms.
The infinite summations over the complete set of unperturbed atomic states appearing in the Nth-order time-dependent perturbation theory are performed using the Green's function formalism. In order to test the method extended calculations of the cross section for two- and three-photon ionization for atomic hydrogen are made and numerical comparison is shown with the values obtained by different methods. The resonant character of the process is made evident since the poles of the radial part of the Green's function represent the actual energy spectrum of the atom. The relevant transition matrix elements are written in closed form and then computed exactly without averaging over the intermediate states. The cross sections vanish for circularly polarized light near some energies which explains the form of the cross sections for circularly and linearly polarized light.
We report total cross-section (TCS) results for low-energy positron scattering from the noble gas xenon. A comparison with previous measurements shows a good level of accord with the recent results of the ANU group (2011 New J. Phys. 13 125004). Very good qualitative agreement is also found with the convergent close-coupling (CCC) calculation of Fursa and Bray (2012 New J. Phys. 14 035002) over most of the common energies. By using the shape of the CCC results as a guide, we also extrapolate our measured cross sections to very low energy. With the aid of the CCC theory, we therefore derive the first experimental estimate for the positron–xenon scattering length of a = −99.2 ± 18.4 au. This value is found to be consistent with the CCC-based estimate and also with those of some other theories. This result supports the existence of a positron–xenon virtual state at a positron energy ε = (1.4 ± 0.6) × 10 −3 eV.