The WKB Method for a Complex Potential

Abstract

SummaryThe extension of the WKB method to a complex potential, as used in the optical "model of the nucleus, is discussed. The formula for the complex phase shifts is formally -deduced, and its accuracy tested against exact calculations for a square potential well and a well with sloping sides. At low energies there occur large discrepancies; the WKB phases vary regularly with energy, whereas the exact values oscillate violently .about the WKB values in a characteristic way and marked resonances occur. The factors affecting the accuracy of the method are discussed.At higher energies the fluctuation of the phases about the WKB value is less marked, and its effect largely cancels out as the result of a larger number of phases being involved in the scattering.1. INTRODUCTION The WKB method for the determination of phase shifts in collision problems has been remarkably useful, notably in the scattering of electrons by atoms (Massey and Burhop 1952), and its range of validity is well understood. With the success of the optical model of the nucleus, in which a potential with an imaginary component is used, there arises the question of the correct method of application of the WKB method to a complex potential and its range of validity.The method has been found valid for nucleons incident on nuclei with energies large compared with the nuclear potential (Mohr and Robson 1956), in which case one may use an" obvious approximation to the WKB formula (Massey and Mohr 1934). In this approximation the ratio of the imaginary to the real component of the phase is equal to the ratio of the imaginary to the real component of the well depth.At lower energies, in a calculation of IX-particle scattering with a comparatively large imaginary well depth, B. A. Robson (unpublished data) in this laboratory recently found that a tentative adaption of the WKB formula without approximation gave an imaginary component of the phase many times the value given by numerical integration of the wave equation.

Cross sections of neutron-induced reactions

“…The phase shift δ in a WKB approximation is [ K dr − k dr] and the real part of it to a zeroth-order approximation [10] for a square well with radius R is (K − k)R, where K and k are the real parts of K and k , respectively. The real wave numbers inside and outside the nucleus are, therefore, given by K 2 = 2m(E + V )/h 2 and k 2 = 2mE/h 2 , respectively.…”

Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase

“…We are now ready to write down the effective commutative time dependent Schrödinger equation in quantum space-time by taking the representation of (48) in |x, t) basis. Using (43,44,47) we finally get,…”

“…3 where we have made use of the coherent state representation of the phase space variables, which are given by (43,44,47) [11],…”

Theoretical estimates of cross sections for neutron–nucleus collisions

“…The optical potential for nuclear n-N interaction can be written as −V − iW with V and W as positive quantities, and contains no Coulomb interaction. The phase shift δ in a WKB approximation is [ K ′ dr − k ′ dr] and the real part of it to a zeroth order approximation (Mohr 1957) for a square well with radius R is (K − k)R where K is the real part of K ′ and k ′ = k is real due to absence of potential. The real wave numbers inside and outside the nucleus are, therefore, given by K 2 = 2m(E + V )/h 2 and k 2 = 2mE/h 2 respectively.…”