1000 Solved Problems in Modern Physics

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 by

σ=

(

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.