Pauli Representation

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

$$\chi_\pm \equiv \chi_{1/2,\pm 1/2}.$$ (734)

It thus follows, from Eqs. (717) and (718), that

$$S_z \chi_\pm = \pm \frac{1}{2} \hbar \chi_\pm,$$ (735)

$$S^2 \chi_\pm = \frac{3}{4} \hbar^2 \chi_\pm.$$ (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

$$\chi_+^\dag \chi_+ = \chi_-^\dag \chi_- = 1,$$ (737)

and

$$\chi_+^\dag \chi_- = 0.$$ (738)

A general spin state can be represented as a linear combination of $\chi_+$ and $\chi_-$: i.e.,

$$\chi = c_+ \chi_+ + c_- \chi_-.$$ (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.,

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

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

$$\chi^\dag\equiv (c_+^\ast, c_-^\ast).$$ (741)

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

$$\chi^\dag \chi = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}... ...+ + c_-^\ast c_- = \vert c_+\vert^2 + \vert c_-\vert^2\geq 0.$$ (742)

Likewise, the product $\chi^\dag \chi'$ of two different spin states is also obtained from the rules of matrix multiplication: i.e.,

$$\chi^\dag \chi' = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+'\ c_-'\end{array}\right) = c_+^\ast c_+' + c_-^\ast c_-'.$$ (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.,

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

$$A^\dag\equiv \left(\begin{array}{cc}A_{11}^\ast,& A_{21}^\ast\\ A_{12}^\ast,& A_{22}^\ast\end{array}\right).$$ (745)

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

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

and

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

$$S_i = \frac{\hbar}{2} \sigma_i.$$ (748)

Here, the $\sigma_i$ are dimensionless $2\times 2$ matrices. According to Eqs. (702)-(704), the $\sigma_i$ satisfy the commutation relations

$$[\sigma_x, \sigma_y] = 2 {\rm i} \sigma_z,$$ (749)

$$[\sigma_y, \sigma_z] = 2 {\rm i} \sigma_x,$$ (750)

$$[\sigma_z,\sigma_x] = 2 {\rm i} \sigma_y.$$ (751)

Furthermore, Eq. (735) yields

$$\sigma_z \chi_\pm = \pm \chi_\pm.$$ (752)

It is easily demonstrated, from the above expressions, that the $\sigma_i$ are represented by the following matrices:

$$\sigma_x \equiv \left(\begin{array}{cc}0,&1\\ 1,& 0\end{array}\right),$$ (753)

$$\sigma_y \equiv \left(\begin{array}{cc}0,&-{\rm i}\\ {\rm i},& 0\end{array}\right),$$ (754)

$$\sigma_z \equiv \left(\begin{array}{cc}1,&0\\ 0,& -1\end{array}\right).$$ (755)

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

Finally, a general spinor takes the form

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

If the spinor is properly normalized then

$$\chi^\dag \chi = \vert c_+\vert^2 + \vert c_-\vert^2 =1.$$ (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 $-\hbar/2$.