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Pauli Spin Matrices
∗
I.
The Pauli spin matrices are
Sx =
¯h
Sy =
¯h
0 −i
i 0
Sz =
¯h
but we will work with their unitless equivalents
σx =
σy =
0 −i
i 0
σz =
where we will be using this matrix language to discuss a spin 1/2 particle.
We note the following construct:
σxσy − σy σx =
0 −i
i 0
0 −i
i 0
0 K
II.
σxσy − σy σx =
0 −i
i 0
0 −i
i 0
which is
σxσy − σy σx =
i 0
0 −i
i 0
0 −i
which is, finally,
σxσy − σy σx =
2 i 0
0 − 2 i
= 2iσK
III.
σxσy − σy σx =
2 i 0
0 − 2 i
= 2iσz
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We can do the same again,
σxσz − σz σx =
which is
σxσz − σz σx =
which is, finally,
σxσz − σz σx =
= − 2 iσy
Summarizing, we have
[σx, σy ] = 2iσz
[σy , σz ] = 2iσx
and, by cyclic permutation.
[σz , σx] = 2iσK
IV.
[σz , σx] = 2iσy
Next, we compute σ
2 i.e.,
σ
2 = σ
2 x +^ σ
2 y +^ σ
2 z = ( 0 1
1 0
0 −i
i 0
0 −i
i 0
σ
2 = σ
2 x +^ σ
2 y +^ σ
2 z = ( 1 0
0 1
0 K
V.
σ
2 = σ
2 x
2 y
2 z
We need the commutator of σ
2 with each component of σ. We obtain
[σ
2 , σx] =
i 0
0 i
i 0
0 i
with the same results for σy and σz , since σ 2 is diagonal. Since the three components of spin individually do not
commute, i.e., [σx, σy ] 6 = 0 as an example, we know that the three components of spin can not simultaneously be
measured. A choice must be made as to what we will simultaneously measure, and the traditional choice is σ 2 and
σz. This is analogous to the L
2 and Lz choice made in angular momentum.
IX.
σz β = − 1 β
We note in passing that
σxα =
= β
X.
It is appropriate to form ladder operators, just as we did with angular momentum, i.e.,
σ
= σx + ıσy
and
σ
− = σx − ıσy
which in matrix form would be
σ
=
0 −ı
ı 0
Clearly
σ
β = Kα
XI.
σ
β = 2α
and
σ
α = 0
as expected. Similar results for the down ladder operator follow immediately.
σ
−
− ı
0 −ı
ı 0
Clearly
σ
− α =?β
We need to observe a particularly strange behaviour of spin operators (and their matrix representatives.
σxσy + σy σx =
0 −i
i 0
0 −i
i 0
which is
( i 0
0 −i
−i 0
0 i
→ K
XII.
i 0
0 −i
−i 0
0 i
This is known as “anti-commuatation”, i.e., not only do the spin operators not commute amongst themselves, but the
anticommute! They are strange beasts.
XIII.
With 2 spin systems we enter a different world. Let’s make a table of possible values:
spin 1 spin 2 denoted as
1/2 1/2 α(1)α(2)
1/2 -1/2 α(1)β(2)
-1/2 1/2 β(1)α(2)
-1/2 -1/2 β(1)β(2)
It makes sense to construct some kind of “ 4-dimensional” representation for this double spin system, i.e.,
α(1)α(2) →
α(1)β(2) →
β(1)α(2) →
β(1)β(2) →
These are the “unit vectors” in the space of interest. Each unit vector stands for a meaningful combination of the
spins. It is sometimes shorter to drop the (1) and (2) and just agree that the left hand designator points to spin-
and the right hand one to spin-2.
Summarizing, in all the relevant notations, we have
as this would operate on α(1)α(2) and generate the correct result.
Σxαα =
Remember, < i|Σx|j > must be evaluated 16 times in our case (less if we recognize the symmetries).
We need to work through all the four basis vectors to obtain the complete representation of ~Σ. We have
Σx =
One can understand each term by writing, as an example,
< 2 |Σx| 1 >= (0, 1 , 0 , 0) ⊗ Σx ⊗
which would be
< 2 |Σx| 1 >= α(1)β(2) ⊗ Σx ⊗ (α(1)α(2))
which is
< 2 |Σx| 1 >= α(1)β(2) ⊗ (β(1)α(2) + α(1)β(2)) = 1
Similarly we obtain
Σy =
0 −ı −ı 0
ı 0 0 −ı
ı 0 0 −ı
0 ı ı 0
and, finally,
Σz =
It is interesting to form ~Σ · ~Σ, i.e.,
2 = Σ
2 x + Σ
2 y + Σ
2 z =
2
0 −ı −ı 0
ı 0 0 −ı
ı 0 0 −ı
0 ı ı 0
2
2
which is
2
which is, finally
2
This last result is called “block diagonal”, and consists of a juxtaposition of a 1x1 matrix, followed by a 2x2 followed
by another 1x1 matrix. This property shows its “ugly/beautiful” head again often, especially in group theory.
It is apparent that α(1)α(2) is an eigenfunction of Σ
2 , i.e.,
2
and simultaneously, α(1)α(2) is an eigenfunction of Σz :
This means that α(1)α(2) is an observable state of the system ( as is β(1)β(2)). Notice further that neither α(1)β(2)
nor β(1)α(2) is an eigenfunction of either Σ 2 or Σz. Instead, linear combinations of these two states are appropriate,
i.e.,
2 (α(1)β(2) + β(1)α(1)) =
where the bracketing has to be studied to see that we are adding the two column vectors before multiplying from the
left with the spin operator. The result is
which shows that the functions α(1)β(2) + α(2)β(1) are eigenfunctions of Σ 2 as expected.
The other linear combination, α(1)β(2) − β(1)α(2) works in the same manner.
2 (α(1)β(2) − β(1)α(1)) =
= zero ⊗
and
Σz (α(1)β(2) − β(1)α(2)) =
= zero ×
in normalized form. Juxtaposing these four eigenvectors we obtain a matrix, T, of the form
T =
√^1 2
√^1 2
1 √ 2
1 √ 2
and, “spinning” (pun, pun, pun) around the main diagonal, we have
T
†
√^1 2
√^1 2
1 √ 2
1 √ 2
such that that the construct T
† S
2 opT^ is
T
† S
2 opT^ =
1 √ 2
1 √ 2
√^1 2
√^1 2
1 √ 2
1 √ 2
√^1 2
√^1 2
which is
T
† S
2 opT^ =
1 √ 2
1 √ 2
√^1 2
√^1 2
8 √ 2
√^8 2
which becomes
T
† S
2 opT^ =
The conjoined eigenvectors constructed to make the matrix T, create a matrix which, when operating on the S
2 op
matrix representative of S 2 in the manner indicated, diagonalizes it. The composite operations are known as a
similarity transformation.
