P307/P707 - Nuclei and Particle Physics
Assignment 1
Submission deadline : 27.01.25
1. Consider the Hamiltonian of a two dimensional isotropic harmonic oscillator
H=1
2(p2
x+p2
y) + 1
2(x2+y2)(1)
What are the wave functions and energies of the three lowest levels? Introduce the perturbation
V=1
2 xy(x2+y2)where 1.(2)
Calculate the first order correction to the energy for the small perturbation. 10
2. Consider an electron to be confined in the interior of a hollow spherical cavity of radius
R
.
Consider the walls of the cavity impenetrable. Compute the pressure exerted on the walls by
the electron in its ground state. 10
3. (a) Assume that the angular dependence is known for the differential cross-section for
Rutherford scattering, i.e.
dσ
dΩ∝ sin θ
2!−4
.(3)
Following dimensional analysis, reconstruct the overall constant and dependencies on the
charge, energy etc. Provide reasonable argument to support your reconstruction.
(b) Consider a proton beam of 10
12
particles per second and 200 MeV/c momentum. The
beam passes through an aluminium sheet of 0.01 cm thickness. Density of aluminium is 2.7
gm/cm
3
. Compute the Rutherford differential cross-section for this beam at an angle
θ
= 30
o
.
(c) Compute the integrated scattering cross-section for angles greater than 5o.
(d) How many protons are scattered out of the beam into the angles >5oper second? 20
4. A beam of 400 MeV electrons are scattered by Gold nuclei through an angle of 15
o
.
Compute the momentum transfer and the Mott cross-section. 10
5. (a) Estimate the nuclear density in CGS unit.
(b) Considering that the neutron star has the similar density as a nucleus and similar mass as
the sun, compute its radius. 10