Chemistry Reference
In-Depth Information
is the nuclear magneton and g
N
¼
5
585. Connected to the nuclear spin
equivalent of (5.13) is the nuclear magnetic resonance (NMR), an experi-
mental technique of great importance nowadays in structural organic
chemistry.
:
5.2 THE PAULI EQUATIONS FOR
ONE-ELECTRON SPIN
Until now nothing has been said about the x and y components of the
vector operator
S
. If we introduce the spin ladder operators
S
þ
¼
S
x
þ
iS
y
;
S
¼
S
x
iS
y
ð
5
:
15
Þ
(i being the imaginary unit, i
2
¼
1),
3
then the analogy with orbital
angular momentum suggests for spin the two-step ladder of Figure 5.1,
where the operators S
þ
and S
respectively step-up (raise) or step-down
(lower) the spin functions upon which they act. Therefore, it is intuitive
that
(
S
þ
a ¼
0
ð
top of the ladder
Þ;
S
a ¼ b
ð
5
:
16
Þ
S
þ
b ¼ a;
S
b ¼
0
ð
bottom of the ladder
Þ
By adding and subtracting the corresponding equations, it is easily seen
that the two spin states
a
and
b
do satisfy the equations
<
1
2
b;
1
2
i
1
2
a
S
x
a ¼
S
y
a ¼
S
z
a ¼
b;
ð
5
:
17
Þ
1
2
a;
1
2
i
1
2
b
:
S
x
b ¼
S
y
b ¼
S
z
b ¼
a;
which are known as Pauli's equations for the one-electron spin and which
are of fundamental importance for thewhole theory of spin, even inmany-
electron systems.
Since
2
2
x
þ
S
2
y
þ
S
2
z
S
¼ S S ¼
S
ð
5
:
18
Þ
S
þ
and S
have the commutation properties
½
S
z
;
S
þ
¼
S
þ
; ½
S
z
;
S
¼
S
; ½
S
2
;
S
¼
0.
3
