By D.S.F. Crothers

This textual content issues ‘semiclassical’ inside of a number of meanings. those contain the time-honored JWKB approximation and its phase-integral generalizations in bankruptcy 2 to 2 and 4 transition issues without or with one or poles: by way of corollary, crossing and non-crossing nonadiabatic collision thought. Above and under threshold Wannier ionization is roofed in bankruptcy three the place the massive parameters are the inverses of the difference of the hyperspherical angles from their ridge values. The extra frequent influence parameter remedy, within which the potentially relativistic heavy-particle relative movement is handled classically and the electrons quantally, is definitely lined in bankruptcy four. Diffusion in solids and beverages is defined in bankruptcy five the place mostly the massive parameter is the peak of the barrier that's conquer by means of thermal agitation. Hypergeometric capabilities are brought in bankruptcy 1 and Mittag-Leffler features in Appendix B.

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78). 87) to be unity. 116) In the Born approximation f (r) is unity. 120) 38 2 Semiclassical Phase Integrals where 01 is the excitation energy. 123) g(q) = exp(−iq · r)V10 (r)d3 r a summation or average over states diﬀering only in magnetic quantum number being assumed [158]. 122) only in that the upper limit to the integration is k0 (∞) + k1 (∞) instead of being infinite. The eﬀect of the latter diﬀerence is minute unless extremely close to the threshold. This represents a remarkable success for the forced-common-turning-point version of the semiclassical treatment.

This approximation corresponds to an expansion in powers of . 6). 1 Approximation 23 where k2 (x0 ) = 0 = k12 (x0 ) ⇒ x0 is a turning point. We expect these equations to be valid if S 1 /S 0 is small. 16) where k(c) = 1. 18) Hence the fractional change in k over the distance λ/4π should be small. This is a small-wavelength or high-frequency approximation. In reality this is the inverse of perturbation theory, that is, the basic quantity is large rather than small. 18) is violated, to the extent that the left-hand side = +∞ (k (x0 ) 0).

85 eV above threshold. 157) gave 3744(πa20 ). 157) substantially diﬀered, that is, until the individual transition probabilities depended on significant contributions from other than the crossing point. 149) becomes more diﬃcult as α increases. 157) respectively, thus showing that the common-turning-point method predicts the wave distortion very accurately for all internuclear separations. 149) it is 285. 149) respectively, are somewhat out of phase for all l, but this is not too surprising in view of the very rapid oscillations of Q01 l as a function of l (cf.