Power Series

$\displaystyle H$	$\displaystyle =$	$\displaystyle T_c \sigma_x,\quad \sigma_x \equiv\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}\right)$
$\displaystyle \sigma_x^0$	$\displaystyle =$	$\displaystyle \hat{1},\quad \sigma_x^1 = \sigma_x,\quad \sigma_x^2 = \hat{1}$
	$\displaystyle \leadsto$	$\displaystyle e^{\alpha \sigma_x} = \hat{1} + \frac{\alpha}{1!}\sigma_x + \frac... ...a^2}{2!}\hat{1} + \frac{\alpha^3}{3!}\sigma_x + \frac{\alpha^4}{4!}\hat{1} +...$
	$\displaystyle =$	$\displaystyle \cosh(\alpha) \hat{1} + \sinh (\alpha) \sigma_x$
$\displaystyle \leadsto U(t,t_0)$	$\displaystyle \equiv$	$\displaystyle e^{-i H (t-t_0)}= \cosh(-i(t-t_0)T_c) \hat{1} + \sinh (-i(t-t_0)T_c) \sigma_x$
	$\displaystyle =$	$\displaystyle \cos [(t-t_0)T_c] \hat{1} -i \sin [(t-t_0)T_c] \sigma_x.$	(1.12)