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
±
:=

S
z
,
±
,
and define
S
x
=
¯
h
2

+

+

+

S
y
=
i
¯
h
2

+
  
+

S
z
=
¯
h
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
,
±
are in fact the eigenvectors of
S
z
with eigenvalues
±
¯
h/
2. Labeling matrix elements in this basis as
A
ij
=
+

A

+
+

A


A

+

A

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

i
i
0
(
S
z
)
ij
=
¯
h
2
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
,
±
=
1
√
2
(

+
± 
)
,

S
y
,
±
=
1
√
2
(

+
±
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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Physics 6210/Spring 2007/Lecture 4
Spectral decomposition
The spin operators have the general form
A
=
ij
A
ij

i
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
not hard to see that this result is quite general. If

i
is an ON basis of eigenvectors of
A
with eigenvalues
a
i
:
A

i
=
a
i

i ,
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 Spring '07
 M
 Physics, Linear Algebra, mechanics, Hilbert space, Sx, sz, Selfadjoint operator

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