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Pauli Representation

Let us denote the two independent spin eigenstates of an electron as
\chi_\pm \equiv \chi_{1/2,\pm 1/2}.
\end{displaymath} (734)

It thus follows, from Eqs. (717) and (718), that
$\displaystyle S_z \chi_\pm$ $\textstyle =$ $\displaystyle \pm \frac{1}{2} \hbar \chi_\pm,$ (735)
$\displaystyle S^2 \chi_\pm$ $\textstyle =$ $\displaystyle \frac{3}{4} \hbar^2 \chi_\pm.$ (736)

Note that $\chi_+$ corresponds to an electron whose spin angular momentum vector has a positive component along the $z$-axis. Loosely speaking, we could say that the spin vector points in the $+z$-direction (or its spin is ``up''). Likewise, $\chi_-$ corresponds to an electron whose spin points in the $-z$-direction (or whose spin is ``down''). These two eigenstates satisfy the orthonormality requirements
\chi_+^\dag \chi_+ = \chi_-^\dag \chi_- = 1,
\end{displaymath} (737)

\chi_+^\dag \chi_- = 0.
\end{displaymath} (738)

A general spin state can be represented as a linear combination of $\chi_+$ and $\chi_-$: i.e.,
\chi = c_+ \chi_+ + c_- \chi_-.
\end{displaymath} (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.,

\chi \equiv \left(\begin{array}{c}c_+\ c_-\end{array}\right).
\end{displaymath} (740)

The corresponding dual vector is represented as a row vector: i.e.,
\chi^\dag\equiv (c_+^\ast, c_-^\ast).
\end{displaymath} (741)

Furthermore, the product $\chi^\dag \chi$ is obtained according to the ordinary rules of matrix multiplication: i.e.,
\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} (742)

Likewise, the product $\chi^\dag \chi'$ of two different spin states is also obtained from the rules of matrix multiplication: i.e.,
\chi^\dag \chi' = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+'\ c_-'\end{array}\right) = c_+^\ast c_+' + c_-^\ast c_-'.
\end{displaymath} (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 $A$ is represented as a $2\times 2$ matrix which operates on a spinor: i.e.,

A \chi \equiv \left(\begin{array}{cc}A_{11},& A_{12}\\
...rray}\right)\left(\begin{array}{c}c_+\ c_-\end{array}\right).
\end{displaymath} (744)

As is easily demonstrated, the Hermitian conjugate of $A$ is represented by the transposed complex conjugate of the matrix used to represent $A$: i.e.,
A^\dag\equiv \left(\begin{array}{cc}A_{11}^\ast,& A_{21}^\ast\\
A_{12}^\ast,& A_{22}^\ast\end{array}\right).
\end{displaymath} (745)

Let us represent the spin eigenstates $\chi_+$ and $\chi_-$ as

\chi_+ \equiv \left(\begin{array}{c}1\ 0\end{array}\right),
\end{displaymath} (746)

\chi_- \equiv \left(\begin{array}{c}0\ 1\end{array}\right),
\end{displaymath} (747)

respectively. Note that these forms automatically satisfy the orthonormality constraints (737) and (738). It is convenient to write the spin operators $S_i$ (where $i=1,2,3$ corresponds to $x,y,z$) as
S_i = \frac{\hbar}{2} \sigma_i.
\end{displaymath} (748)

Here, the $\sigma_i$ are dimensionless $2\times 2$ matrices. According to Eqs. (702)-(704), the $\sigma_i$ satisfy the commutation relations
$\displaystyle [\sigma_x, \sigma_y]$ $\textstyle =$ $\displaystyle 2 {\rm i} \sigma_z,$ (749)
$\displaystyle [\sigma_y, \sigma_z]$ $\textstyle =$ $\displaystyle 2 {\rm i} \sigma_x,$ (750)
$\displaystyle [\sigma_z,\sigma_x]$ $\textstyle =$ $\displaystyle 2 {\rm i} \sigma_y.$ (751)

Furthermore, Eq. (735) yields
\sigma_z \chi_\pm = \pm \chi_\pm.
\end{displaymath} (752)

It is easily demonstrated, from the above expressions, that the $\sigma_i$ are represented by the following matrices:
$\displaystyle \sigma_x$ $\textstyle \equiv$ $\displaystyle \left(\begin{array}{cc}0,&1\\
1,& 0\end{array}\right),$ (753)
$\displaystyle \sigma_y$ $\textstyle \equiv$ $\displaystyle \left(\begin{array}{cc}0,&-{\rm i}\\
{\rm i},& 0\end{array}\right),$ (754)
$\displaystyle \sigma_z$ $\textstyle \equiv$ $\displaystyle \left(\begin{array}{cc}1,&0\\
0,& -1\end{array}\right).$ (755)

Incidentally, these matrices are generally known as the Pauli matrices.

Finally, a general spinor takes the form

\chi = c_+ \chi_++c_- \chi_- = \left(\begin{array}{c}c_+\ c_-\end{array}\right).
\end{displaymath} (756)

If the spinor is properly normalized then
\chi^\dag \chi = \vert c_+\vert^2 + \vert c_-\vert^2 =1.
\end{displaymath} (757)

In this case, we can interpret $\vert c_+\vert^2$ as the probability that an observation of $S_z$ will yield the result $+\hbar/2$, and $\vert c_-\vert^2$ as the probability that an observation of $S_z$ will yield the result $-\hbar/2$.

next up previous
Next: Spin Precession Up: Spin Angular Momentum Previous: Eigenstates of and
Richard Fitzpatrick 2010-07-20