Document 2347267

Qm 3 Hw5
1. The Helium atom consists of 2 electrons each of spin ½ .
In this question the interaction between the electrons may be related as a perturbation .
a) Using the assumption that the Helium nuclei is very heavy , find the Hamiltonian for the
electrons
b) What are the stable bound states of the nuclei, neglect the interaction between the
electrons
c) What is the ionization energy of Helium atom, consider zeroth order correction
d) What is the 1st order correction to the ionization energy of Helium nuclei
e) Find the leading order correction to the bound states energy
• In part (4) the integrals look similar to those in electrostatic problem
• In part (5) find closed expressions but you don’t have to calculate the integrals
• Pay attention in last part you have to analyze two cases- singlet and triplet
2.
An angular momentum eigen-state
j, m j
is rotated by the infinitesimal angle
ε about
the y-axis.
Obtain the expression for the probability for the new rotated state to be found in the original rotated
state up to terms of
ε2.
3.
The motion of an electron in a central field is described by a Hamiltonian of the form
H
= H 0 + H so
P2
=
H so ζ ( r ) L ⋅ S . The spin-orbit coupling leads to energy differences
where H
=
+ V ( r ) and
0
2m
2
2
2
between levels with the same values of L , S but different values of J where J=L+S.
a) Find the commutation relations between [H,L2], [H,S] , [H,LZ] and [H,SZ], [H,J2], [H,JZ].
b) Consider the stationary states of H that are also eigen-states of the operators that commute with
the Hamiltonian. Express the angular part of these eigen-functions in terms of spherical harmonics and
two-component spinors .
c) Let the eigen-functions of part (b) be characterised by the quantum numbers l,j,m (which are related
to the eigen values of L2, J2, Jz respectively). Determine the possible values of Lz and Sz and find their
probabilities and average values.
4. Parity in 3 dimensions
The parity operator in 3 dimensions is defined as