48
Physics Formulary by ir. J.C.A. Wevers
10.10
Spin
For the spin operators are defined by their commutation relations:
[
S
x
, S
y
] =
i
¯
hS
z
. Because the spin operators
do not act in the physical space
(
x, y, z
)
the uniqueness of the wavefunction is not a criterium here: also half
oddinteger values are allowed for the spin. Because
[
L, S
] = 0
spin and angular momentum operators do not
have a common set of eigenfunctions. The spin operators are given by
S
=
1
2
¯
h
σ
, with
σ
x
=
0
1
1
0
,
σ
y
=
0

i
i
0
,
σ
z
=
1
0
0

1
The eigenstates of
S
z
are called
spinors
:
χ
=
α
+
χ
+
+
α

χ

, where
χ
+
= (1
,
0)
represents the state with
spin up (
S
z
=
1
2
¯
h
) and
χ

= (0
,
1)
represents the state with spin down (
S
z
=

1
2
¯
h
). Then the probability
to find spin up after a measurement is given by

α
+

2
and the chance to find spin down is given by

α


2
. Of
course holds

α
+

2
+

α


2
= 1
.
The electron will have an intrinsic magnetic dipole moment
M
due to its spin, given by
M
=

eg
S
S/
2
m
,
with
g
S
= 2(1 +
α
/
2
π
+
· · ·
)
the gyromagnetic ratio. In the presence of an external magnetic field this gives
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 Spring '10
 Ye
 Physics, Electron, Dirac equation, Spin Operators, J.C.A. Wevers

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