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Physical Review A

Comment on the accuracy of the uniform WKB theory of inelastic collisions

Hervé Guérin, Thomas P. Tsien, Byung Chan Eu, Ronald E. Olson

Published Feb 1, 1974• 7 citations
Publisher

Abstract

In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the ${\mathrm{He}}^{+}$-Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the ${\mathrm{He}}^{+}$-Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant ${a}_{4}$ by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-St\"uckelberg theory, and is found to behave correctly in the strong-coupling regime as ${a}_{4}$ increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-St\"uckelberg theory over all values of ${a}_{4}$ for the system studied.

DOI
10.1103/physreva.9.995

Related papers

Improved solutions to the equation of motion in the uniform WKB theory for two-channel problems

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The equation of motion in the uniform WKB theory for a two-channel problem is reinvestigated in order to obtain improved approximate solutions that would perform better than the first order Magnus approximation solution previously used in the analysis of two-channel inelastic collision problems. We have developed two approximate solutions and investigated their variants in the paper. The two main improved approximate solutions are obtained by properly resumming the Magnus expansion in a novel way and also by appropriately modifying the equation of motion such that the Magnus approximation used on it would not damage the accuracy of the solution thus obtained. The solutions obtained give rise to transition probabilities well comparable over a wide range of the coupling constant with those obtained by the numerical solution of the coupled Schrödinger equations.

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Theory of inelastic scattering of a particle in the near-adiabatic limit

Lit-Deh Chang, Walter Kohn

A semiclassical theory for inelastic scattering of a particle by a two-state system in the near-adiabatic limit is developed. The exponentially small transition amplitude is calculated. The theory is an extension of Pokrovskii and Khalatnikov's theory for above-barrier reflection (Zh. Eksp. Teor. Fiz. 40, 1713 (1961) [Sov. Phys.---JETP 13, 1207 (1961)]). The analysis involves studies of the WKB solutions in the complex coordinate plane along certain contours. The region of validity of the theory is established. Our result has the same form as the Landau-Zener-Stueckelberg formula; however, our theory is applicable to more general systems. Numerical comparisons with exact solutions are presented; the differences between our results and exact solutions become negligible as the adiabatic limit is approached.

Physical Review A•Aug 1, 1987•0 citations
In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the He+{\mathrm{He}}^{+}He+-Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the He+{\mathrm{He}}^{+}He+-Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant a4{a}_{4}a4​ by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-St"uckelberg theory, and is found to behave correctly in the strong-coupling regime as a4{a}_{4}a4​ increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-St"uckelberg theory over all values of a4{a}_{4}a4​ for the system studied.
The equation of motion in the uniform WKB theory for a two-channel problem is reinvestigated in order to obtain improved approximate solutions that would perform better than the first order Magnus approximation solution previously used in the analysis of two-channel inelastic collision problems. We have developed two approximate solutions and investigated their variants in the paper. The two main improved approximate solutions are obtained by properly resumming the Magnus expansion in a novel way and also by appropriately modifying the equation of motion such that the Magnus approximation used on it would not damage the accuracy of the solution thus obtained. The solutions obtained give rise to transition probabilities well comparable over a wide range of the coupling constant with those obtained by the numerical solution of the coupled Schrödinger equations.
For pt.I see ibid., vol.14, p.985 (1981). A general theory of the Rydberg atom inelastic scattering by perturber atoms has been developed previously, in which the leading term is the simple impulse amplitude, followed by the various correction terms. Theoretical estimates of these correction terms are presented in this paper, resulting in a general scaling formula. The non-impulse correction to the impulse approximation varies as n -2 at large principal quantum number n and depends on the signature of the electron-perturber scattering length. The distortion correction due to the interaction between the perturber and the Rydberg atomic core is large for the intermediate values of n<or approximately=20 behaves as n -4 , and depends on the perturber-core total cross section. The scaling formula derived here has been tested for the scattering systems consisting of Na and Rb targets and inert gas perturbers.
Different theoretical approaches to the famous two-state Landau-Zener problem are briefly discussed. Apart from traditional methods of the adiabatic perturbation theory, the Born-Oppenheimer approximation with geometric phase effects, the two-level approach, and the momentum space representation, the problem is treated semiclassically in the coordinate space. In the framework of the instanton approach, we present a full and unified description of the 1D Landau-Zener problem of level crossing. The method enables us to accurately treat all four transition points (appearing at two-level crossing), while the standard WKB approach takes only two of them into account. The latter approximation is adequate for calculating the transition probability or for studying scattering processes, but it does not work in finding the corresponding chemical reaction rates, in which all four transition points can often be relevant in the typical range of parameters. Applications of the method and of the results may concern various systems in physics, chemistry, and biology.
WKB theory provides a plausible link between classical mechanics and quantum mechanics in its semi-classical limit. Connecting the WKB wave function across the turning points in a quantum well, leads to the WKB quantization condition. In this work, I focus on some improvements and recent developments related to the WKB quantization condition. First I discuss how the combination of super-symmetric quantum mechanics and WKB, gives the SWKB quantization condition which is exact for a large class of potentials called shape invariant potentials. Next I turn to the fact that there is always a probability of refection when the potential is not constant and the phase of the wave function should account for this refection. WKB theory ignores refection except at turning points. I explain the work of Friedrich and Trost who showed that by including the correct "refection phase" at a turning point, the WKB quantization condition can be made to give exact bound state energies. Next I discuss the work of Cao and collaborators which takes refection of the wave function into account everywhere. We show that Cao's method provides a way to compute the F-T refection phase. Finally I discuss a paper of Fabre and Guery-Odelin who used the exponential potential to study the accuracy of WKB. In their results the accuracy deteriorates as the energy increases, which is inconsistent with Bohr's correspondence principle. Using the Friedrich and Trost method, we resolved this problem.
A semiclassical theory for inelastic scattering of a particle by a two-state system in the near-adiabatic limit is developed. The exponentially small transition amplitude is calculated. The theory is an extension of Pokrovskii and Khalatnikov's theory for above-barrier reflection (Zh. Eksp. Teor. Fiz. 40, 1713 (1961) [Sov. Phys.---JETP 13, 1207 (1961)]). The analysis involves studies of the WKB solutions in the complex coordinate plane along certain contours. The region of validity of the theory is established. Our result has the same form as the Landau-Zener-Stueckelberg formula; however, our theory is applicable to more general systems. Numerical comparisons with exact solutions are presented; the differences between our results and exact solutions become negligible as the adiabatic limit is approached.