Exercise 4.5

Consider \(\hat{n} = (n_x, n_y, n_z)\) is a real unit vector in three dimension, and \(\vec{\sigma} = (X, Y, Z)\). Then

(1)\[\begin{split} \begin{align} (\hat{n}\cdot \vec{\sigma} )^{2} =& (n_xX + n_yY + n_zZ)(n_xX + n_yY + n_zZ) \\ =& n_x^2 X^2 + n_xn_yXY + n_xn_zXZ + n_xn_yYX \\ &+ n^2_yY^2 + n_yn_zYZ+ n_xn_zZX + n_yn_zZY + n_z^{2}Z^{2}\\ =& n_x^2 I + n_xn_yXY + n_xn_zXZ + n_xn_yYX \\ &+ n^2_yI + n_yn_zYZ+ n_xn_zZX + n_yn_zZY + n_z^{2}I\\ \end{align} \end{split}\]

Notice that for Pauli matrices, we have the anti-commutation rule as

(2)\[ \{X, Y\} = \{Y, Z\} = \{X, Z\} =0 \]

Then we could simplify eq. (1) as

(3)\[ \begin{align} (\hat{n}\cdot \vec{\sigma} )^{2} =& n_x^2 I + n^2_yI + n_z^{2}I = I \end{align} \]

where we have \(n^2_x + n^2_y + n^2_z = 1\) for unit vector \(\hat{n}\). From Exercise 4.2 we have \(\exp (iAx) = \cos (x)I + i\sin(x)A\) where \(A^2 = I\). Let \(A = \hat{n}\cdot\vec{\sigma}\) and \(x= -\theta/2\), we have

(4)\[\begin{split} \begin{align} \exp (-i\hat{n}\cdot\vec{\sigma}\theta/2) &= \cos \left(\frac{\theta}{2}\right)I - i\sin\left(\frac{\theta}{2}\right)(\hat{n}\cdot\vec{\sigma}) \\ &= \cos \frac{\theta}{2}I - i\sin\frac{\theta}{2}(n_xX + n_yY + n_zZ) \\ \end{align} \end{split}\]