3.2 Problems 153
3.102 A particle of mass m is trapped in a potential well which has the form,
(V=^1 /^2 mω^2 x^2. Use the variation method with the normalized trial function
1 /√
a)
cos(πx/ 2 a) in the limits−a<x<a, to find the best value ofa.3.103 In Problem 3.45, consider the perturbationW(x, y)=W 0 for 0<x<a/ 2
and 0<y<a/2, and 0 elsewhere. Calculate the first order perturbation
energy.
3.2.8 Scattering (Phase-Shift Analysis) .....................
3.104 A beam of particles of energy^2 k^2 / 2 m,movinginthe+zdirection, is scat-
tered by a short-range central potentialV(r). One looks for the stationary
solution of the Schrodinger equation which is of the asymptotic form,
ψ≈eikz+f(θ)eikr/r
Derive the partial-wave decomposition
f(θ)=(2ik)−^1∑∞
l= 0(2l+1) (exp(2iδl)−1)Pl(cosθ)[Adapted from the University College, Dublin, Ireland, 1967]3.105 In the case ofα−Hescattering the measured scattered intensity at 45◦
(laboratory coordinates) is twice the classical result. Indicate how the wave-
mechanical theory of collisions explains this experimental result.
[Adapted from the University of New Castle 1964]
3.106 In the analysis of scattering of particles of massmand energyEfrom a fixed
centre with rangea, the phase shift for thelth partial wave is given by
δl=sin−^1[
(iak)l
[( 2 l+ 1 )!(l!)]^1 /^2]
Show that the total cross-section at a given energy is approximately given byσ=(
2 π^2
mE)
exp(
−
2 mEa^2
^2)
[University of Cambridge, Tripos]3.107 At what neutron lab energy willp-wave be important inn–pscattering?
3.108 1 MeV neutrons are scattered on a target. The angular distribution of the neu-
trons in the centre-of-mass is found to be isotropic and the total cross-section
is measured to be 0. 1 b. Using the partial wave representation, calculate the
phase shifts of the partial waves involved.
3.109 Considering the scattering from a hard sphere of radius a such that onlys-
andp-waves are involved, the potential being
V(r)=∞forr<a
=0forr>a.