25  1
5.73
Lecture
#25
revised 4 November, 2002
H
SO
+ H
Zeeman
Coupled vs. Uncoupled Basis Sets
Last time:
matrices for
J
2
,
J
+
,
J
–
,
J
z
,
J
x
,
J
y
in
jm
j
⟩
basis for J = 0, 1/2, 1
Pauli spin 1/2 matrices
arbitrary
2 × 2
When
M
is
ρ
→
visualization of fictitious vector in fictitious Bfield
When
M
is a term in
H
→
idea that arbitrary operator can be
decomposed as sum of
J
i
.
TODAY:
1.
H
SO
+
H
Zeeman
as illustrative
2.
Dimension of basis sets
JLSM
J
⟩
and
LM
L
SM
S
⟩
is same
3.
matrix elements of
H
SO
in both basis sets
4.
matrix elements of
H
Zeeman
in both basis sets
5.
ladders and orthogonality for transformation between basis
sets.
Necessary to be able to evaluate matrix elements of
H
Zeeman
in coupled basis.
Why?
Because coupled basis set
does not explicitly give effects of
L
z
or
S
z
.
M
I
=
+
⋅
a
a
0
1
r
r
σ
types of operators
e.g. magnetic moment (
is a known constant or a function of r)
how to evaluate matrix elements (e.g. Stark Effect)
e.g. Spin  Orbit
a
a
J
q
J
J
r
1
2
⋅
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25  2
5.73
Lecture
#25
revised 4 November, 2002
1
2
2
0
.
( )
H
s
s
H
s
s
SO
n
Zeeman
z
z
z
z
z
r
B
=
⋅
≡
⋅
= −
+
(
)
≡ −
(
)
+
(
)
ξ
ζ
γ
ω
Suppose we have 2 kinds of angular momenta, which can be coupled to each other to
form a
total
angular momentum.
The components of
L,S,
and
J
each follow the standard angular momentum
commutation rule, but
These commutation rules specify that
L
and
S
act like vectors wrt
J
but as scalars wrt
to each other.
Coupled
uncoupled
vs.
m
representations.
j sm
sm
j
s
*
matrix elements of certain operators are more convenient in one basis set than the
other
*
a unitary transformation between basis sets must exist
*
limiting cases for energy level patterns
will give a factor of
anomalous g  value of
e
−
r
r
r
J
L
S
→
→
→
jm
m
sm
j
s
and
s
will
each give a
factor of
ζ
ω
n
and
are in rad / s
0
(
)
r r
L S
J L
L
J S
S
,
,
,
,
.
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 Spring '04
 RobertField
 Angular Momentum, Trigraph, Basis set, basis sets, uncoupled basis, B. Uncoupled Representation

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