WKBJ approximations
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1: 33.23 Methods of Computation
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§33.23(vii) WKBJ Approximations
►WKBJ approximations (§2.7(iii)) for are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. … ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for and in the region inside the turning point: .2: 34.8 Approximations for Large Parameters
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►Semiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7).
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3: 2.7 Differential Equations
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§2.7(iii) Liouville–Green (WKBJ) Approximation
… ►The first of these references includes extensions to complex variables and reversions for zeros. …4: 2.9 Difference Equations
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§2.9(iii) Other Approximations
►For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999). …5: 33.22 Particle Scattering and Atomic and Molecular Spectra
6: 9.16 Physical Applications
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►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other.
The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood.
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►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions.
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►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point.
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