Skip to main content

# 9.4: Pauli Representation

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

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)

$$\newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}$$ $$\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}$$ $$\newcommand {\btau}{\mbox{\boldmath\tau}}$$ $$\newcommand {\bmu}{\mbox{\boldmath\mu}}$$ $$\newcommand {\bsigma}{\mbox{\boldmath\sigma}}$$ $$\newcommand {\bOmega}{\mbox{\boldmath\Omega}}$$ $$\newcommand {\bomega}{\mbox{\boldmath\omega}}$$ $$\newcommand {\bepsilon}{\mbox{\boldmath\epsilon}}$$

This page titled 9.4: Pauli Representation is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

• Was this article helpful?