Download 2/25/11 QUANTUM MECHANICS II (524) PROBLEM SET 6 (hand in

2/25/11
QUANTUM MECHANICS II (524)
PROBLEM SET 6 (hand in March 4)
21) Consider a deuterium atom (composed of a nucleus with spin I = 1 and an
electron). The electron angular momentum is denoted by J = L + S, where L
is the orbital angular momentum of the electron and S its spin. The total
angular momentum of the atom is F = J + I, where I is the nuclear spin.
a) What are the possible values of the quantum numbers J and F for a
deuterium atom in the 1s ground state?
b) Same questions for deuterium in the 2p excited state.
22) (20 points) The hydrogen atom nucleus is a proton with spin I = 1/2.
a) In the notation of the preceding problem, what are the possible values of
the quantum numbers J and F for a hydrogen atom in the 2p level?
b) Use the notation {|n`mi} for the eigenstates of the “simple” hydrogen
Hamiltonian studied last semester. When we add the electron spin to its
orbital angular momentum, we consider states coupled to the total electron
angular momentum {|n`s = 1/2JMJ i}. Finally, the states that have
coupled J and I to F are denoted by {|n`sJIF MF i}. The magnetic
moment operator of the electron is
M = µB (L + 2S)/~.
In each of the subspaces arising from the 2p level corresponding to fixed
values of J and F , the projection theorem enables us to write
M = gJF µB F /~.
Calculate the various possible values of the Landé factors gJF
corresponding to the 2p level.
23) a) Wave-vector space wave functions for free particles with good quantum
numbers for the energy, orbital angular momentum and its projection are
denoted by hk|E`mi. We have seen that this wave function can be written
as
1/2
4π
hk|E`mi = f`E (k)
Y`m (k̂)
2` + 1
and that
~2 k 2
).
2m
Determine N up to a phase choice by employing the normalization
f`E (k) = N δ(E −
hE 0 `0 m0 |E`mi = δ(E 0 − E)δ`0 ` δm0 m .
b) Use the integral representation of the spherical Bessel function (Eq. (127)
on p. 131 in the book) to obtain Eq. (129).