9.4: Pauli Representation

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

\[\chi_\pm \equiv \chi_{1/2,\pm 1/2}. \nonumber \]

It thus follows, from Equations \ref{e10.16} and \ref{e10.17}, that

\[\begin{align} S_z\,\chi_\pm &= \pm \frac{1}{2}\,\hbar\,\chi_\pm,\label{e10.34}\\[4pt] S^2\,\chi_\pm &= \frac{3}{4}\,\hbar^2\,\chi_\pm.\end{align} \]

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

\[ \chi_+^\dagger\,\chi_+ = \chi_-^\dagger\,\chi_- = 1, \label{e10.35} \]

and

\[ \chi_+^\dagger\,\chi_- = 0. \label{e10.36} \]

A general spin state can be represented as a linear combination of \(\chi_+\) and \(\chi_-\): that is,

\[\chi = c_+\,\chi_+ + c_-\,\chi_-. \nonumber \]

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: that is,

\[\chi \equiv \left(\begin{array}{c}c_+\\c_-\end{array}\right). \nonumber \]

The corresponding dual vector is represented as a row vector: that is,

\[\chi^\dagger\equiv (c_+^\ast, c_-^\ast). \nonumber \]

Furthermore, the product \(\chi^\dagger\,\chi\) is obtained according to the ordinary rules of matrix multiplication: that is,

\[\chi^\dagger\,\chi = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+\\c_-\end{array}\right) = c_+^\ast\,c_+ + c_-^\ast\,c_- = |c_+|^2 + |c_-|^2\geq 0. \nonumber \]

Likewise, the product \(\chi^\dagger\,\chi'\) of two different spin states is also obtained from the rules of matrix multiplication: that is,

\[\chi^\dagger\,\chi' = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+'\\c_-'\end{array}\right) = c_+^\ast\,c_+' + c_-^\ast\,c_-'. \nonumber \]

Note that this particular representation of spin space is in complete accordance with the discussion in Section 1.3. 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: that is,

\[A\,\chi \equiv \left(\begin{array}{cc}A_{11},& A_{12}\\ A_{21},& A_{22}\end{array}\right)\left(\begin{array}{c}c_+\\c_-\end{array}\right). \nonumber \]

As is easily demonstrated, the Hermitian conjugate of \(A\) is represented by the transposed complex conjugate of the matrix used to represent \(A\): that is,

\[A^\dagger \equiv \left(\begin{array}{cc}A_{11}^\ast,& A_{21}^\ast\\ A_{12}^\ast,& A_{22}^\ast\end{array}\right). \nonumber \]

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

\[\chi_+ \equiv \left(\begin{array}{c}1\\0\end{array}\right), \nonumber \]

and

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

respectively. Note that these forms automatically satisfy the orthonormality constraints \ref{e10.35} and \ref{e10.36}. It is convenient to write the spin operators \(S_i\) (where \(i=1,2,3\) corresponds to \(x,y,z\)) as

\[ S_i = \frac{\hbar}{2}\,\sigma_i. \label{e10.46} \]

Here, the \(\sigma_i\) are dimensionless \(2\times 2\) matrices. According to Equations \ref{e10.1x}–\ref{e10.2x}, the \(\sigma_i\) satisfy the commutation relations

\[\begin{align} &= 2\,{\rm i}\,\sigma_z,\\[4pt] [\sigma_y, \sigma_z]&= 2\,{\rm i}\,\sigma_x,\\[4pt] [\sigma_z,\sigma_x]&= 2\,{\rm i}\,\sigma_y.\end{align} \nonumber \]

Furthermore, Equation \ref{e10.34} yields

\[\sigma_z\,\chi_\pm = \pm \chi_\pm. \nonumber \]

It is easily demonstrated, from the previous expressions, that the \(\sigma_i\) are represented by the following matrices:

\[\begin{align} \sigma_x&\equiv \left(\begin{array}{cc}0,&1\\ 1,& 0\end{array}\right),\\[4pt] \sigma_y&\equiv \left(\begin{array}{cc}0,&-{\rm i}\\ {\rm i},& 0\end{array}\right),\\[4pt] \sigma_z&\equiv \left(\begin{array}{cc}1,&0\\ 0,& -1\end{array}\right).\label{e10.53}\end{align} \]

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

Finally, a general spinor takes the form

\[\chi = c_+\,\chi_++c_-\,\chi_- = \left(\begin{array}{c}c_+\\c_-\end{array}\right). \nonumber \]

If the spinor is properly normalized then

\[\chi^\dagger\,\chi = |c_+|^2 + |c_-|^2 =1. \nonumber \]

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