Absorption of Ultra-Sonic Waves by Hydrogen and Carbon by Abello T.P.

By Abello T.P.

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The analytical properties of the coupling and energy difference ∆W in this a When Λ (R) correlate to the same atomic orbital in the relevant two electronic states En 2 the united atom limit R → 0, ∆E(R) ∝ R and the electronic matrix element of L+ (or L− ) in Eq. 61) is not equal to zero at R = 0. 4. 15:22 9in x 6in Nonadiabatic Transition: Concepts, . . 1. 37 Analytical properties of the various nonadiabatic coupling schemes. Potential energy difference ∆E ∝ Coupling scheme Coupling T ∝ Adiabatic-state representation (R − R∗ )−1 (R − R∗ )1/2 Radial (R∗ :complex) rotational R2 (a) Degeneracy at R = 0 (b) Crossing at finite R = Rx R − RX (c) No crossing Constant R−2 Dynamical-state representation (R − R∗ )−1 (R − R∗ )1/2 Any transition (R∗ :complex) representation are the same as those of the original radial coupling problems; and thus the semiclassical theories developed for the latter can now be applied in a unified way to any transitions in the new representation.

Stokes and anti-Stokes lines in the case of Airy function. - - -: Stokes line, ——: anti-Stokes line. (Taken from Ref. ) rules, the solution given by Eq. 24) in region 1 of Fig. 25) region 6 : −i(A + BT1 )(z, ·)d − i[B + T2 (A + BT1 )](·, z)s region 7 : −i(A + BT1 )(z, ·)s − i[B + T2 (A + BT1 )](·, z)d region 1 : −i{(A + BT1 ) + T3 [B + T2 (A + BT1 )]}(z, ·)s −i[B + T2 (A + BT1 )](·, z)d . Since the last equation of Eq. 25) should coincide with the first one of Eq. 25) for abritrary A and B, we can easily obtain T 1 = T2 = T3 = i .

1. Wentzel–Kramers–Brillouin Semiclassical Theory If we could know such wavefunctions in the whole range of coordinate space that satisfy necessary physical boundary conditions, then we could solve all the corresponding physical problems completely. This is not usually the case, however, and we definitely need approximate analytical wavefunctions in order to formulate basic physical problems. Such approximate analytical wave functions are provided by the Wentzel–Kramers–Brillouin approximation in the case of potential problem.

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