By Boris Mikhailovich Karnakov, Vladimir Pavlovich Krainov
This e-book has developed from lectures dedicated to purposes of the Wentzel - Kramers – Brillouin- (WKB or quasi-classical) approximation and of the tactic of 1/N −expansion for fixing numerous difficulties in atomic and nuclear physics. The purpose of this booklet is to aid scholars and investigators during this box to increase their wisdom of those vital calculation equipment in quantum mechanics. a lot fabric is contained herein that isn't to be chanced on somewhere else. WKB approximation, whereas constituting a basic quarter in atomic physics, has now not been the point of interest of many books. a singular technique has been followed for the presentation of the subject material, the fabric is gifted as a succession of difficulties, by means of a close method of fixing them. The equipment brought are then used to calculate Rydberg states in atomic structures and to guage strength boundaries and quasistationary states. eventually, adiabatic transition and ionization of quantum platforms are coated.
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Extra resources for WKB Approximation in Atomic Physics
12) Hnr (ξ) exp −ξ 2 /2 . √ Here once more notation is introduced ξ ≡ nω (r − r0 ) . According to Eq. 10), the unperturbed harmonic oscillator has the frequency ω0 = nω. In the region of localization of a particle the typical deviations of this particle from the equilibrium point are small: −1/2 (r − r0 ) ∼ ω0 ∼ n −1/2 1. Analogous estimates are valid for deviations of radial coordinate in anharmonic terms which can be considered as small perturbation of the harmonic oscillator. We write this perturbation in the form of sum of two terms: one term is even function of the variable x ≡ r − r0 while the other term is odd function of this variable V = V (+) + V (−) ; V (+) (x) = 2r12 nr (nr + 1) − 23 n (2nr + 1) 0 V (−) (x) = (2nr + 1) n rx3 + n 2 αx 3 .
However, we should substitute g by g0 and put ε(0) = 0. Hence, we obtain g0 = − 1 ; r0 v (r0 ) = −2v(r0 ). 36) (1) The first approximation can be found from Eq. 12) by formal substitution εnr → −g1 v(r0 ). Taking into account Eq. 36), we obtain g1 = − 2nr + 1 1 ω− 2 2v(r0 ) r0 = (2nr + 1) ωr02 − 1 g0 . 37) Here, the frequency ω is given by Eq. 11), but with substitution g → g0 . 13)). Taking it into account, we obtain g2 = 15 22 ω 3 + s A2 − (2 + s) B v (r0 ) 16 15 8 √ 1+s 1 s 6A C + 5C + 1 − − C+ 8 4 C .
The condition of applicability of WKB approximation is fulfilled for Rydberg states on the radial dimensions r aB = 2 m e Z e2 . 4) for electron energy allows to neglect the dependence of the quantum defect on the principal quantum number n. 1 Obtain the energy spectrum, Eq. 1) of Rydberg states of atoms using WKB wave function, Eq. 3). 3) corresponds to taking into account the boundary condition at r → 0 in the exact radial wave function, where Eq. 3) is inapplicable. The boundary condition in the infinity, r → ∞, is Rnl (r ) → 0.
WKB Approximation in Atomic Physics by Boris Mikhailovich Karnakov, Vladimir Pavlovich Krainov