# Additional WKB Inversion Relations for Bound‐State and Scattering Problems

15 May 1971-Journal of Chemical Physics (American Institute of PhysicsAIP)-Vol. 54, Iss: 10, pp 4174-4177

Abstract: New inversion relations for the diatom system (elastic scattering of two atoms by a single potential, or a bound diatomic molecule) are presented. They are exact within the WKB approximation for the relevant quantities and are closely related to the well‐known RKR expressions. Explicit formulas are given for bound‐state and scattering input.

Topics: WKB approximation (58%), Elastic scattering (54%), Scattering (53%), Bound state (52%), Diatomic molecule (51%)

## Summary (1 min read)

### II. Inversion from Eigenvalues

- The principal disadvantage o~ Equation (9) is that one cannot solve these equations to obtain e)..'":plicit e>.:pressions for r< and r> (due to the logarithms).
- For bound-state problems there is no apparent reason >vhy any pair of equa.tions should be easier to use than the usual RKR pair [Equations (10) and (11) ].
- Equations (12) and (13) do give alternate inversionrelations, however, v.rhich may be of use in some situations.

### b. Curve Crossing Probability

- In the semiclassical treatment 1 7 of' electronic transitions in atom-atom collisions which take place via a crossing of the two potentj.al curves, the transition probability is the product of a slm-rly varying factor and an oscillatory factor.
- The phase of the oscillatory fae;tor is a phase integral; more specifically, it is the difference in phase integrals on the initial and final potential curves from their respective classical turning points to the crossing point.
- If the E and £ dependence of this phase integral can be extracted from the differential and/or total cross section for the electronic transition, then the inversion formulae of Section II can be utilized to construct the two potentials involved.

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##### References

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01 Jan 1966

Abstract: Much progress has been made in scattering theory since the publication of the first edition of this book fifteen years ago, and it is time to update it. Needless to say, it was impossible to incorporate all areas of new develop- ment. Since among the newer books on scattering theory there are three excellent volumes that treat the subject from a much more abstract mathe- matical point of view (Lax and Phillips on electromagnetic scattering, Amrein, Jauch and Sinha, and Reed and Simon on quantum scattering), I have refrained from adding material concerning the abundant new mathe- matical results on time-dependent formulations of scattering theory. The only exception is Dollard's beautiful "scattering into cones" method that connects the physically intuitive and mathematically clean wave-packet description to experimentally accessible scattering rates in a much more satisfactory manner than the older procedure. Areas that have been substantially augmented are the analysis of the three-dimensional Schrodinger equation for non central potentials (in Chapter 10), the general approach to multiparticle reaction theory (in Chapter 16), the specific treatment of three-particle scattering (in Chapter 17), and inverse scattering (in Chapter 20). The additions to Chapter 16 include an introduction to the two-Hilbert space approach, as well as a derivation of general scattering-rate formulas. Chapter 17 now contains a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect.

3,911 citations

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01 Jan 1949

Abstract: Volume II of this work covers many-body problems and applications of the theory to electron collisions with atoms, collisions between atomic systems, nuclear collisions, certain aspects of two-body systems under relativistic collisions, and the use of time-dependent perturbation theory. Despite the amount of work carried out since this book was first published, the underlying theory presented here remains both sound and of practical value to all theoretical physicists.

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119 citations

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Abstract: A simple, physically intuitive, expression has been obtained for the scattering phase shift δ (l, E) in the case that the effective radial potential possesses a maximum. The result [Eq. (3)] is seen to modify the usual WKB phase shift [Eq. (5)] by replacing a step function by a particular “smooth step function.” On the basis of this result, resonances in the energy dependence of the total elastic cross section are discussed. From qualitative arguments it is seen that metastable states more than ∼0.6 ℏωB energy units above the top of the barrier are too short lived to have physical significance (ωB is the harmonic frequency related to the inverted barrier). Application is also made to the eigenvalues of a hindered rotor.

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