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Pauli Representation
Let us denote the two independent spin eigenstates of an electron as
![\begin{displaymath}
\chi_\pm \equiv \chi_{1/2,\pm 1/2}.
\end{displaymath}](img1777.png) |
(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
![\begin{displaymath}
\chi_+^\dag \chi_+ = \chi_-^\dag \chi_- = 1,
\end{displaymath}](img1786.png) |
(737) |
and
![\begin{displaymath}
\chi_+^\dag \chi_- = 0.
\end{displaymath}](img1787.png) |
(738) |
A general spin state can be represented as a linear combination of
and
: i.e.,
![\begin{displaymath}
\chi = c_+ \chi_+ + c_- \chi_-.
\end{displaymath}](img1788.png) |
(739) |
It is thus evident that electron spin space is two-dimensional.
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 so-called 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 two-dimensional space: i.e.,
![\begin{displaymath}
\chi \equiv \left(\begin{array}{c}c_+\ c_-\end{array}\right).
\end{displaymath}](img1789.png) |
(740) |
The corresponding dual vector is represented as a row vector: i.e.,
![\begin{displaymath}
\chi^\dag\equiv (c_+^\ast, c_-^\ast).
\end{displaymath}](img1790.png) |
(741) |
Furthermore, the product
is obtained according to the
ordinary rules of matrix multiplication: i.e.,
![\begin{displaymath}
\chi^\dag \chi = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}...
...+ + c_-^\ast c_- = \vert c_+\vert^2 + \vert c_-\vert^2\geq 0.
\end{displaymath}](img1791.png) |
(742) |
Likewise, the product
of two different spin states
is also obtained from the rules of matrix multiplication: i.e.,
![\begin{displaymath}
\chi^\dag \chi' = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+'\ c_-'\end{array}\right) = c_+^\ast c_+' + c_-^\ast c_-'.
\end{displaymath}](img1792.png) |
(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.,
![\begin{displaymath}
A \chi \equiv \left(\begin{array}{cc}A_{11},& A_{12}\\
A_{...
...rray}\right)\left(\begin{array}{c}c_+\ c_-\end{array}\right).
\end{displaymath}](img1794.png) |
(744) |
As is easily demonstrated, the Hermitian conjugate of
is represented by
the transposed complex conjugate of the matrix used to represent
: i.e.,
![\begin{displaymath}
A^\dag\equiv \left(\begin{array}{cc}A_{11}^\ast,& A_{21}^\ast\\
A_{12}^\ast,& A_{22}^\ast\end{array}\right).
\end{displaymath}](img1795.png) |
(745) |
Let us represent the spin eigenstates
and
as
![\begin{displaymath}
\chi_+ \equiv \left(\begin{array}{c}1\ 0\end{array}\right),
\end{displaymath}](img1796.png) |
(746) |
and
![\begin{displaymath}
\chi_- \equiv \left(\begin{array}{c}0\ 1\end{array}\right),
\end{displaymath}](img1797.png) |
(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
![\begin{displaymath}
S_i = \frac{\hbar}{2} \sigma_i.
\end{displaymath}](img1800.png) |
(748) |
Here, the
are dimensionless
matrices. According
to Eqs. (702)-(704), the
satisfy the commutation
relations
Furthermore, Eq. (735) yields
![\begin{displaymath}
\sigma_z \chi_\pm = \pm \chi_\pm.
\end{displaymath}](img1808.png) |
(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
![\begin{displaymath}
\chi = c_+ \chi_++c_- \chi_- = \left(\begin{array}{c}c_+\ c_-\end{array}\right).
\end{displaymath}](img1815.png) |
(756) |
If the spinor is properly normalized then
![\begin{displaymath}
\chi^\dag \chi = \vert c_+\vert^2 + \vert c_-\vert^2 =1.
\end{displaymath}](img1816.png) |
(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
2010-07-20