$$\require{cancel}$$

# 9.4: Pauli Representation

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

$\chi_\pm \equiv \chi_{1/2,\pm 1/2}.$ It thus follows, from Equations ([e10.16]) and ([e10.17]), that \begin{aligned} S_z\,\chi_\pm &= \pm \frac{1}{2}\,\hbar\,\chi_\pm,\label{e10.34}\\[0.5ex] S^2\,\chi_\pm &= \frac{3}{4}\,\hbar^{\,2}\,\chi_\pm.\end{aligned} 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

$\label{e10.35} \chi_+^\dagger\,\chi_+ = \chi_-^\dagger\,\chi_- = 1,$ and

$\label{e10.36} \chi_+^\dagger\,\chi_- = 0.$ A general spin state can be represented as a linear combination of $$\chi_+$$ and $$\chi_-$$: that is, $\chi = c_+\,\chi_+ + c_-\,\chi_-.$ 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, $\chi \equiv \left(\begin{array}{c}c_+\\c_-\end{array}\right).$ The corresponding dual vector is represented as a row vector: that is, $\chi^\dagger\equiv (c_+^\ast, c_-^\ast).$ Furthermore, the product $$\chi^\dagger\,\chi$$ is obtained according to the ordinary rules of matrix multiplication: that is, $\chi^\dagger\,\chi = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+\\c_-\end{array}\right) = c_+^\ast\,c_+ + c_-^\ast\,c_- = |c_+|^{\,2} + |c_-|^{\,2}\geq 0.$ Likewise, the product $$\chi^\dagger\,\chi'$$ of two different spin states is also obtained from the rules of matrix multiplication: that is, $\chi^\dagger\,\chi' = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+'\\c_-'\end{array}\right) = c_+^\ast\,c_+' + c_-^\ast\,c_-'.$ 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, $A\,\chi \equiv \left(\begin{array}{cc}A_{11},& A_{12}\\ A_{21},& A_{22}\end{array}\right)\left(\begin{array}{c}c_+\\c_-\end{array}\right).$ 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, $A^\dagger \equiv \left(\begin{array}{cc}A_{11}^\ast,& A_{21}^\ast\\ A_{12}^\ast,& A_{22}^\ast\end{array}\right).$

Let us represent the spin eigenstates $$\chi_+$$ and $$\chi_-$$ as $\chi_+ \equiv \left(\begin{array}{c}1\\0\end{array}\right),$ and $\chi_- \equiv \left(\begin{array}{c}0\\1\end{array}\right),$ respectively. Note that these forms automatically satisfy the orthonormality constraints ([e10.35]) and ([e10.36]). It is convenient to write the spin operators $$S_i$$ (where $$i=1,2,3$$ corresponds to $$x,y,z$$) as

$\label{e10.46} S_i = \frac{\hbar}{2}\,\sigma_i.$ Here, the $$\sigma_i$$ are dimensionless $$2\times 2$$ matrices. According to Equations ([e10.1x])–([e10.2x]), the $$\sigma_i$$ satisfy the commutation relations \begin{aligned} [\sigma_x, \sigma_y]&= 2\,{\rm i}\,\sigma_z,\\[0.5ex] [\sigma_y, \sigma_z]&= 2\,{\rm i}\,\sigma_x,\\[0.5ex] [\sigma_z,\sigma_x]&= 2\,{\rm i}\,\sigma_y.\end{aligned} Furthermore, Equation ([e10.34]) yields $\sigma_z\,\chi_\pm = \pm \chi_\pm.$ It is easily demonstrated, from the previous expressions, that the $$\sigma_i$$ are represented by the following matrices: \begin{aligned} \sigma_x&\equiv \left(\begin{array}{cc}0,&1\\ 1,& 0\end{array}\right),\\[0.5ex] \sigma_y&\equiv \left(\begin{array}{cc}0,&-{\rm i}\\ {\rm i},& 0\end{array}\right),\\[0.5ex] \sigma_z&\equiv \left(\begin{array}{cc}1,&0\\ 0,& -1\end{array}\right).\label{e10.53}\end{aligned} Incidentally, these matrices are generally known as the Pauli matrices.

Finally, a general spinor takes the form $\chi = c_+\,\chi_++c_-\,\chi_- = \left(\begin{array}{c}c_+\\c_-\end{array}\right).$ If the spinor is properly normalized then $\chi^\dagger\,\chi = |c_+|^{\,2} + |c_-|^{\,2} =1.$ 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 $$-\hbar/2$$.

# Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
