By Boris Mikhailovich Karnakov, Vladimir Pavlovich Krainov
This ebook has developed from lectures dedicated to functions of the Wentzel - Kramers – Brillouin- (WKB or quasi-classical) approximation and of the strategy of 1/N −expansion for fixing quite a few difficulties in atomic and nuclear physics. The purpose of this booklet is to assist scholars and investigators during this box to increase their wisdom of those vital calculation tools in quantum mechanics. a lot fabric is contained herein that's not to be came upon in different places. WKB approximation, whereas constituting a primary region in atomic physics, has no longer been the point of interest of many books. a singular strategy has been followed for the presentation of the subject material, the fabric is gifted as a succession of difficulties, by means of an in depth means of fixing them. The equipment brought are then used to calculate Rydberg states in atomic platforms and to guage capability boundaries and quasistationary states. eventually, adiabatic transition and ionization of quantum structures are coated.
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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 .
As an example, we consider the most unfavorable case l = 0 for 1/n-expansion. The exact energy spectrum of a particle in Hulthén potential with l = 0 is given by exact =− E nn r 2 2m R 2 U0 1 8m R 2 (nr + 1)2 2 2 − (nr + 1)2 , l = 0. 30) Using Eq. 27), we obtain exact E nn =− r 2 8m R 2 (nr + 1)2 2 (exp(r0 ) − 1)2 −1 r03 exp(r0 ) 2 . 31) Here n = nr + 1 for states with l = 0. 6 the results of calculations of energies for ground 1s−state (nr = 0) using Eqs. 31) at the various values of potential parameters.
42 2 1/N-Expansion in Quantum Mechanics U (0)eff (r) 2 1 E (0)nn(2) (0) Fig. 1 Dependence of the effective potential Ueff = n 2 gv (r ) + 1/2r 2 (see Eq. 16 It is seen that 1/n-expansion is applicable as well as at n = 1. 5. The depth of the potential well diminishes with increase of r0 ; the ratio g/gcr illustrates this fact. , when a new level appears (see Fig. 1). According to Eq. 5 for the ground level. 1/n-expansion is inapplicable for shallow s-levels, since the wave function of s−level is localized far outside of the short-range potential at E → 0.