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Pauli Representation
Let us denote the two independent spin eigenstates of an electron as

(734) 
It thus follows, from Eqs. (717) and (718), that
Note that corresponds to an electron whose spin angular momentum vector has a positive component along the axis. Loosely speaking,
we could say that the spin vector points in the direction (or its spin is
``up''). Likewise,
corresponds to an electron whose spin points in the direction
(or whose spin is ``down'').
These two eigenstates satisfy the orthonormality requirements

(737) 
and

(738) 
A general spin state can be represented as a linear combination of
and : i.e.,

(739) 
It is thus evident that electron spin space is twodimensional.
Up to now, we have discussed spin space in rather abstract terms. In the
following, we shall describe a particular representation of electron
spin space due to Pauli. This socalled Pauli representation allows us
to visualize spin space, and also facilitates calculations involving spin.
Let us attempt to represent a general spin state as a complex column vector in some twodimensional space: i.e.,

(740) 
The corresponding dual vector is represented as a row vector: i.e.,

(741) 
Furthermore, the product
is obtained according to the
ordinary rules of matrix multiplication: i.e.,

(742) 
Likewise, the product
of two different spin states
is also obtained from the rules of matrix multiplication: i.e.,

(743) 
Note that this particular representation of spin space is in complete accordance with the discussion in Sect. 10.3. For obvious reasons,
a vector used to represent a spin state is generally known as
spinor.
A general spin operator is represented as a matrix
which operates on a spinor: i.e.,

(744) 
As is easily demonstrated, the Hermitian conjugate of is represented by
the transposed complex conjugate of the matrix used to represent : i.e.,

(745) 
Let us represent the spin eigenstates and as

(746) 
and

(747) 
respectively. Note that these forms automatically
satisfy the orthonormality constraints (737) and (738).
It is convenient to write the spin operators (where corresponds to
) as

(748) 
Here, the are dimensionless matrices. According
to Eqs. (702)(704), the satisfy the commutation
relations
Furthermore, Eq. (735) yields

(752) 
It is easily demonstrated, from the above expressions, that the are represented by the
following matrices:
Incidentally, these matrices are generally known as the Pauli matrices.
Finally, a general spinor takes the form

(756) 
If the spinor is properly normalized then

(757) 
In this case, we can interpret as the probability that
an observation of will yield the result , and
as the probability that an observation of
will yield the result .
Next: Spin Precession
Up: Spin Angular Momentum
Previous: Eigenstates of and
Richard Fitzpatrick
20100720