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48
Physics Formulary by ir. J.C.A. Wevers
10.10
Spin
For the spin operators are defned 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 o± the wave±unction is not a criterium here: also hal±
oddinteger values are allowed ±or the spin. Because
[
L,S
]=0
spin and angular momentum operators do not
have a common set o± eigen±unctions. The spin operators are given by
±
±
S
=
1
2
¯
h
±
±σ
, with
±
x
=
±
01
10
²
,
±
y
=
±
0

i
i
0
²
,
±
z
=
±
0

1
²
The eigenstates o±
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 fnd spin up a±ter a measurement is given by

α
+

2
and the chance to fnd spin down is given by

α


2
. O±
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 o± an external magnetic feld this gives
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
 Spring '10
 Ye
 Physics

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