Orbital Angular Momentum Eigenfunctions
Michael Fowler 1/11/08
Introduction
In the last lecture , we established that the operators
2
,
z
JJ
G
have a common set of eigenkets
,
j m
,
()
22
,1
,
,
,
z
Jjm j
j
jmJjm mjm
=+
=
G
=
,
=
where
j
,
m
are integers or half odd
integers, and we found the matrix elements of
,
+
−
(and hence those of
J
x
,
J
y
) between these
eigenkets. This purely formal structure, therefore, nails down the allowed values of total angular
momentum and of any measured component.
But there are other things we need to know: for
example, how is an electron in a particular angular momentum state in an atom affected by an
external field?
To compute that, we need to know the wave function
.
ψ
If a system has spherical symmetry, such as an electron in the Coulomb field of a hydrogen
nucleus, then the Hamiltonian
H
and the operators
2
,
z
G
have a common set of eigenkets
,,
Ejm
.
The spherically symmetric Hamiltonian is unchanged by rotation, so must commute
with any rotation operator,
[ ]
2
,
0 and
,
0
z
HJ
⎡⎤
=
=
⎣⎦
G
. Recall that
commuting Hermitian
operators can be diagonalized simultaneously—and therefore have a common set of eigenkets.
Fortunately, many systems of interest do have spherical symmetry, at least to a good
approximation, the basic example of course being the hydrogen atom, so the natural set of basis
states is the common eigenkets of energy and angular momentum.
It turns out that even when
the spherical symmetry is broken, the angular momentum eigenkets may still be a useful starting
point, with the symmetry breaking treated using perturbation theory.
Two-Dimensional Models
As a warm-up exercise for the complications of the three-dimensional spherically symmetric
model, it is worth analyzing a two-dimensional
circularly
symmetric model, that is,
()()
( )
2
,
2
Hx
y
x
y
V
xy
x
yEx
y
Mx
y
ψψ
⎛⎞
∂∂
=−
+
+
+
=
⎜⎟
⎝⎠
=
,
.
(In this section, we’ll denote the particle mass by
M
, to avoid confusion with the angular
momentum quantum number
m
– but be warned you are often going to find
m
used for both in
the same discussion!)
The two-dimensional angular momentum operator is
.