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 and Attributions

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

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