Question

Consider scatering by a repulsive δ-shell potential. V(r) = 2-5(r-R), (γ > 0) (a) Set up an equation that determines the s-wave phase shift dg as a function of k (E-h2k2/2m). (b) Assume that γ is very large, γ 》 R-1, k. If tan kR is close e resonance behavior is scattering. Determine approximately the positions of the resonances, keeping terms of order of 1/y. Also obtain an ance width Γ defined ation expression for the r as 1.

Answer 1

" For s - waves l = 0 from schrodinger equation for radial function: - -t^2/2m d^2 u(r)/dr62 +[V(r) + d(d + 1) t^2/2 mr^2] u(r) = E u(r) -t^2/2m d^2 u(r)/d r^2 + t^2v/2m delta(r-R) Mu(r) = E u(r) where u(r) = r R(r) Ampersand r Notequalto R, equation yields: d^2u/dr^2 = -R^2 u(r) where R = squareroot 2mE/t^2 the acceptable solution for equation (2) must vanish at r = 0 and finit at r rightarrow Infinity u(r) ={u_1 (r) = A sin(kr);0 < r < R u_2 (r) = B sin(kr + delta o): r greaterthan R continuity of u(r) at r = R, u_2(R) = u_1 (R) leads to B sin(Kr + delta o) = A sin(kR) 0_n other hand, integrating equation (1) (with l = 0) from r = R -E to r = R + E we get-t^2/2m Integral_R-E^R + E d^2 u(r)/dr^2 dr + t^2r/2m integral_R-E^R + E delta(r-r) u(r) dr = E Integral_R-E^R + E u(r) dr and taking limit epsilon rightarrow 0 we end up with du_2(r)/dr