Set 8 due 14 March
1) Sum rules are quite useful in evaluating electromagnetic matrix elements.
The oldest of these is the Thomas-Reiche-Kuhn sum rule
2m
h
2
m
| m|r|i |2 (Em Ei ) = 3
(1)
(a) [10 points] Derive this rule (use commutators). (b) [10 points]
Set 1 due 24 January
Two check perturbation theory formulas problems and then a real physical
one.
1) [15 points] Consider the two state system
H=(
1
)
2
(1)
with all entries real and 1 = 2 . (a) [4 points] Diagonalize it and nd the exact
energies and eig
Set 4 due 14 February
Most of a comment by Fermi: There are two ways of doing theoretical physics.
One way, and this is the way I prefer, is to have a clear physical picture of
the process that you are calculating. The other is to have a precise and selfc
Set 5 due 21 February
1) [15 points] Consider an idealized laser pulse as a time dependent electric eld
E(t) = z E0 exp(t2 / 2 ). Suppose that the pulse is incident on a hydrogen atom
which is in the 1S state at t = . Neglecting spin, compute the probabil
Set 8 due 14 March
1) Sum rules are quite useful in evaluating electromagnetic matrix elements.
The oldest of these is the Thomas-ReicheKuhn sum rule
Z l<mli)i2(Em - E1) = 3 (1)
(a) [10 points] Derive this rule (use commutators). (b) [10 points] If
Set 7 due 7 March
The midterm will be Friday March 7, in G-2B47 (our classroom) from 7 to
830 PM.
1) [20 points] (a) [ 10 points] We already knew that an electric eld interacts with
an electric dipole through an interaction V = er E. Compute E from A (wit
Set 2 due 31 January
1) [15 points] Consider a two-state system with Hamiltonian
H=(
1
)
2
(1)
As you did in set 1, rst nd the energy eigenvalues and eigenfunctions exactly.
Then, assume that the systems is almost degenerate, that |1 2 | < |.
Show that th
Set 3 due 7 February
1) [15 points] In addition to the usual Zeeman eect, there is a quadratic Zeeman
eect associated with the interaction of an atom with a constant magnetic eld
B,
1
(1)
E = B 2
2
arising from the e2 A2 /(2mc2 ) term in the full electrom
Set 6 due 28 February
The midterm will be Friday March 7, in G-2B47 (our classroom) from 7 to
830 PM.
1) [10 points] Show that if the vector potential has its photon normalization,
A(x, t) = (
2 c2 1/2 i(krt)
h
) [e
+ ei(krt) ]
V
(1)
then the eld momentum