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(π x/2a) in the limits −a < x < a, to find the best value of a.
3.103 In Problem 3.45, consider the perturbation W (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
/2m, moving in the +z direction, is scattered by a short-range central potential V (r ). One looks for the stationary
solution of the Schrodinger equation which is of the asymptotic form,
ψ ≈ e
ikz
+ f (θ )e
ikr
/r
Derive the partial-wave decomposition
f (θ ) = (2ik)
−1
∞
l=0
(2l + 1) (exp(2iδ l ) − 1)P l (cos θ )
[Adapted from the University College, Dublin, Ireland, 1967]
3.105 In the case of α − He scattering the measured scattered intensity at 45
◦
(laboratory coordinates) is twice the classical result. Indicate how the wavemechanical 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 mass m and energy E from a fixed
centre with range a, the phase shift for the lth partial wave is given by
δ l = sin
−1
(iak)
l
[(2l + 1)!(l!)]
1/2
Show that the total cross-section at a given energy is approximately given by
σ =
2π
2
m E
exp
−
2m Ea
2
2
[University of Cambridge, Tripos]
3.107 At what neutron lab energy will p-wave be important in n– p scattering?
3.108 1 MeV neutrons are scattered on a target. The angular distribution of the neutrons 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 only sand p-waves are involved, the potential being
V (r ) = ∞ for r < a
= 0 for r > a.
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(π x/2a) in the limits −a < x < a, to find the best value of a.
3.103 In Problem 3.45, consider the perturbation W (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
/2m, moving in the +z direction, is scattered by a short-range central potential V (r ). One looks for the stationary
solution of the Schrodinger equation which is of the asymptotic form,
ψ ≈ e
ikz
+ f (θ )e
ikr
/r
Derive the partial-wave decomposition
f (θ ) = (2ik)
−1
∞
l=0
(2l + 1) (exp(2iδ l ) − 1)P l (cos θ )
[Adapted from the University College, Dublin, Ireland, 1967]
3.105 In the case of α − He scattering the measured scattered intensity at 45
◦
(laboratory coordinates) is twice the classical result. Indicate how the wavemechanical 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 mass m and energy E from a fixed
centre with range a, the phase shift for the lth partial wave is given by
δ l = sin
−1
(iak)
l
[(2l + 1)!(l!)]
1/2
Show that the total cross-section at a given energy is approximately given by
σ =
2π
2
m E
exp
−
2m Ea
2
2
[University of Cambridge, Tripos]
3.107 At what neutron lab energy will p-wave be important in n– p scattering?
3.108 1 MeV neutrons are scattered on a target. The angular distribution of the neutrons 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 only sand p-waves are involved, the potential being
V (r ) = ∞ for r < a
= 0 for r > a.