Chemistry 120A Problem Set 6
(due March 17, 2010)
1. Problems 643, 644 and 645 in McQuarrie and Simon. These problems lead you
through the textbook’s treatment of orbital magnetization in the hydrogen atom.
You might ﬁnd them helpful as background for Problem 2 below.
2. The orbital motion of the electron in a hydrogen atom produces a magnetic moment
in the
z
direction,
μ
z
=

(
e/
2
m
e
c
)
L
z
,
where

e
and
m
e
are, respectively, the charge and mass of an electron,
c
is the speed
of light, and
L
z
is the diﬀerential operator giving the
z
component of the angular
momentum. The energy of a magnetic moment in a static magnetic ﬁeld,
~
B
=
B
ˆ
z
,
is
E
mag
=

μ
z
B
.
Use ﬁrst order perturbation theory for the energy and calculate the expectation
values for the energy when the hydrogen atom is in the states (
n,‘,m
)=(2,1,1),
(2,1,0), and (2,1,1). Compare your results with the magnetic energies discussed in
lecture.
3. The instantaneous dipole moment of a hydrogen atom is
~μ
=

e~
r
, where
~
r
is the
position of the electron relative to that of the nucleus. The energy of a dipole in an
electric ﬁeld,
~
E
, is
E
elec
=

~μ
·
~
E
.
With this formula in mind, you will consider the Hamiltonian for a hydrogen atom
perturbed by an electric ﬁeld and use ﬁrst order
degenerate
perturbation theory to
estimate the energy levels of this atom.
(a) With the unperturbed hydrogen stationary states

ψ
n,‘,m
i
for (
n,‘,m
) = (1,0,0),
(2,0,0), (2,1,1), (2,1,0) and (2,1,1) determine the corresponding elements of the
5
×
5 Hamiltonian matrix for the hydrogen atom in an electric ﬁeld. Take the
z
axis to be the direction of the electric ﬁeld. (Hint: Most of these matrix
elements are zero, and not much work is required to see why. Do or consider
doing your
θ
and
φ
integrations ﬁrst! In this way, you can quickly identify the
few integrals that actually need to be evaluated explicitly.)
(b) Use ﬁrst order degenerate perturbation theory to calculate the perturbed en
ergy eigenvalues for the ﬁve stationary states of lowest energy.
(c) The splitting of energy levels by a static electric ﬁeld is called the Stark eﬀect.
Draw an energy diagram showing the perturbed energies relative to the two
lowest unperturbed energy levels. Indicate how the Stark eﬀect energy level
splittings depend upon the size of the electric ﬁeld,
E
.
1