Exercise 4.2

Suppose \(A\) is a matrix such that \(A^2 = I\) and \(x\) is a real number. We can use Taylor expansion to calculate \(\exp (iAx)\) as

(1)\[\begin{split} \begin{align} \exp (iAx) &= \sum_{n=0}^{\infty}\frac{(iAx)^n}{n!} = I + iAx+\frac{i^2A^2x^2}{2!} + \frac{i^3A^3x^3}{3!} + \frac{i^4A^4x^4}{4!} + \dotsc \\ \end{align} \end{split}\]

From eq. (1) we can find that,

• If \(n\)​ is even number, we have

  (2)\[\begin{split} \begin{align} n\text{ is even: }&I -\frac{A^2x^2}{2!} + \frac{A^4x^4}{4!} - \frac{A^6x^6}{6!} + \frac{A^8x^8}{8!} - \frac{A^{10}x^{10}}{10!}+\dotsc \\ =& I -\frac{x^2}{2!}I + \frac{x^4}{4!}I - \frac{x^6}{6!}I + \frac{x^8}{8!}I - \frac{x^{10}}{10!}I+\dotsc \end{align} \end{split}\]

• If \(n\) is odd number we have

  (3)\[\begin{split} \begin{align} n\text{ is odd: }&iAx -\frac{iA^3x^3}{3!} + \frac{iA^5x^5}{5!} - \frac{iA^7x^7}{7!} + \frac{iA^9x^9}{9!} - \frac{iA^{11}x^{11}}{11!}+\dotsc \\ =& iAx -\frac{iAx^3}{3!} + \frac{iAx^5}{5!} - \frac{iAx^7}{7!} + \frac{iAx^9}{9!} - \frac{iAx^{11}}{11!}+\dotsc \end{align} \end{split}\]

where we simplify eq. (2) and (3) using \(A^2 = I\)​. Note that

(4)\[\begin{split} \begin{align} \sin x &= \sum^{\infty}_{n=0}\frac{(-1)^{n}}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!}- \frac{x^7}{7!} + \frac{x^9}{9!} - \frac{x^{11}}{11!}+\dotsc \\ \cos x &= \sum^{\infty}_{n=0}\frac{(-1)^{n}}{(2n)!}x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}I + \frac{x^8}{8!}I - \frac{x^{10}}{10!}I+\dotsc \end{align} \end{split}\]

we could rewrite eq. (2) as \(\cos(x) I\) and eq. (3) as \(i\sin(x)A\). Thus, we could re-write eq. (1)as

(5)\[ \exp (iAx) = \cos(x)I+ i\sin(x)A \]

From the textbook, we know that the rotation operator about the \(\hat{x}, \hat{y}, \hat{z}\) axes are defined by

(6)\[ R_x(\theta) = e^{-i\theta X/2}, R_y(\theta) = e^{-i\theta Y/2}, R_z(\theta) = e^{-i\theta Z/2} \]

Let \(x = -\theta/2\) and \(A = X, Y, Z\) for \(R_x(\theta), R_y(\theta), R_z(\theta)\), respectively, we could rewrite the rotation operator as

(7)\[\begin{split} \begin{align} R_x(\theta) &= e^{-i\theta X/2} = \cos\frac{\theta}{2}I - i\sin\frac{\theta}{2} X\\ R_y(\theta) &= e^{-i\theta Y/2}=\cos\frac{\theta}{2}I - i\sin\frac{\theta}{2} Y\\ R_z(\theta) &= e^{-i\theta Z/2} =\cos\frac{\theta}{2}I - i\sin\frac{\theta}{2} Z \end{align} \end{split}\]