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Physics 6210/Spring 2007/Lecture 4
Lecture 4
Relevant sections in text:
§
1.2, 1.3, 1.4
The spin operators
Finally we can discuss the deﬁnition of the spin observables for a spin 1/2 system. We
will do this by giving the expansion of the operators in a particular basis. We denote the
basis vectors representing states in which the
z
component of spin is known by
±i
:=

S
z
,
±i
,
and deﬁne
S
x
=
¯
h
2
±

+
ih
+
ih
+

²
S
y
=
i
¯
h
2
±
ih
+
  
+
ih
²
S
z
=
¯
h
2
±

+
ih
+
  ih
²
.
Note that we have picked a direction, called it
z
, and used the corresponding spin states
for a basis. Of course, any other direction could be chosen as well.
You can now check
that, with the above deﬁnition of
S
z
,
±i
are in fact the eigenvectors of
S
z
with eigenvalues
±
¯
h/
2. Labeling matrix elements in this basis as
A
ij
=
³
h
+

A

+
i h
+

A
i
h
A

+
i h
A
i
´
,
you can also verify the following matrix representations
in the
±i
basis
:
(
S
x
)
ij
=
¯
h
2
³
0
1
1
0
´
(
S
y
)
ij
=
¯
h
2
³
0

i
i
0
´
(
S
z
)
ij
=
¯
h
2
³
1
0
0

1
´
.
Finally, you should check that all three spin operators are selfadjoint. It will be quite
a while before you get a deep understanding of why these particular operators are chosen.
For now, let us just take them as given and see what we can do with them.
As a good exercise you can verify that

S
x
,
±i
=
1
√
2
(

+
i ± i
)
,

S
y
,
±i
=
1
√
2
(

+
i ±
i
i
)
are eigenvectors of
S
x
and
S
y
, respectively, with eigenvalues
±
¯
h/
2. Note that these eigen
vectors are normalized to have norm (“length”) unity. The fact that these eigenvectors
are distinct from those of
S
z
will be dealt with a little later. For now, just note that the
three operators do not share any eigenvectors. Note also that the eigenvalues of the spin
operators are all nondegenerate – exercise.
1
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View Full DocumentPhysics 6210/Spring 2007/Lecture 4
Spectral decomposition
The spin operators have the general form
A
=
X
ij
A
ij

i
ih
j

,
which we discussed earlier. Note, though, that
S
z
has an especially simple,
diagonal
form.
This is because it is being represented by an expansion in a basis of its eigenvectors. It is
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.
 Spring '07
 M
 Physics, mechanics

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