Spinor Rotation Matrices

A general rotation operator in spin space is written

$$T ({\mit\Delta}\phi) = \exp\left(\frac{-{\rm i} \,{\bf S}\cdot{\bf n}\,{\mit\Delta}\varphi}{\hbar}\right),$$ (5.95)

by analogy with Equation (5.24), where ${\bf n}$ is a unit vector pointing along the axis of rotation, and ${\mit\Delta}\varphi$ is the angle of rotation. Here, ${\bf n}$ can be regarded as a trivial position operator. The rotation operator is represented

$$\exp\left(\frac{-{\rm i} \,{\bf S}\cdot{\bf n}\,{\mit\Delta}\varp... ...-{\rm i} \,\mbox{\boldmath$\sigma$}\cdot{\bf n}\,{\mit\Delta}\varphi}{2}\right)$$ (5.96)

in the Pauli scheme. The term on the right-hand side of the previous expression is the exponential of a matrix. This can easily be evaluated using the Taylor series for an exponential, plus the rules

$$( \mbox{\boldmath$\sigma$} \cdot {\bf n})^k = 1 \mbox{for $k$\ even} ,$$ (5.97)

$$( \mbox{\boldmath$\sigma$} \cdot {\bf n})^k = \mbox{\boldmath$\sigma$} \cdot {\bf n} \mbox{for $k$\ odd} .$$ (5.98)

These rules follow trivially from the identity (5.93). Thus, we can write

$$\exp\left(\frac{-{\rm i} \,\mbox{\boldmath$\sigma$}\!\cdot\!{\bf n}\,{\mit\Delta}\varphi}{2}\right) = \left[ 1 - \frac{(\mbox{\boldmath$\sigma$}\cdot {\bf n})^2}{2!}... ...t {\bf n})^4}{4!} \left(\frac{{\mit\Delta}\varphi}{2} \right)^4 + \cdots\right]$$

$$\phantom{=}- {\rm i} \left[ (\mbox{\boldmath$\sigma$}\cdot {\bf n... ...ot {\bf n})^3}{3!} \left(\frac{{\mit\Delta}\varphi}{2}\right)^3 + \cdots\right]$$

$$= \cos({\mit\Delta}\varphi/2)- {\rm i}\,\sin ({\mit\Delta}\varphi/2)\, \mbox{\boldmath$\sigma$} \cdot{\bf n}.$$ (5.99)

The explicit $2\times 2$ form of this matrix is

$$\left(\begin{array}{cc} \cos({\mit\Delta}\varphi/2) - {\rm i}\,n_... ...lta}\varphi/2) + {\rm i}\, n_z \sin({\mit\Delta}\varphi/2) \end{array} \right).$$ (5.100)

Rotation matrices act on spinors in much the same manner as the corresponding rotation operators act on state kets. Thus,

$$\chi' = \exp\left(\frac{-{\rm i} \,\mbox{\boldmath$\sigma$}\cdot{\bf n}\,{\mit\Delta}\varphi}{2}\right) \,\chi,$$ (5.101)

where $\chi'$ denotes the spinor obtained after rotating the spinor $\chi$ an angle ${\mit\Delta}\varphi$ about the axis ${\bf n}$ . The Pauli matrices remain unchanged under rotations. However, the quantity $\chi^\dagger \,\sigma_k \,\chi$ is proportional to the expectation value of $S_k$ [see Equation (5.75)], so we would expect it to transform like a vector under rotation. (See Section 5.4.) In fact, we require

$$(\chi^\dagger \,\sigma_k \,\chi)' \equiv (\chi^\dagger)' \sigma_k\, \chi' = \sum_l R_{k\,l}\, (\chi^\dagger \sigma_l\, \chi),$$ (5.102)

where the $R_{kl}$ are the elements of a conventional rotation matrix [92]. This is easily demonstrated, because

$$\exp\left(\frac{\,{\rm i}\,\sigma_3 \,{\mit\Delta}\varphi}{2}\rig... ...{2}\right) = \sigma_1 \cos{\mit\Delta}\varphi -\sigma_2 \sin{\mit\Delta}\varphi$$ (5.103)

plus all cyclic permutations. The previous expression is the $2\times 2$ matrix analog of

$$\exp\left(\frac{\,{\rm i}\,S_z \,{\mit\Delta}\varphi}{\hbar}\righ... ...}{\hbar}\right) = S_x\, \cos{\mit\Delta}\varphi -S_y\, \sin{\mit\Delta}\varphi.$$ (5.104)

[See Equation (5.30).] Equation (5.103) follows from the Baker-Campbell-Hausdorff lemma, (5.31), which holds for matrices, in addition to operators.