# Constructing a time evolution operator $e^{it H}$ for $H^2=I$

Consider a Hamiltonian $$H = \sigma_x \otimes \sigma_z$$ Construct the time evolution operator $$U(t) = \mathrm{e}^{-\frac{iHt}{\frac{h}{2\pi}}}$$ [Hint:Write down the expansion of $$\mathrm{e}^x$$ and use the property of $$H^2$$]

This was one of my assignment problems and I really couldn't make sense of what the hint implied and ended up getting $$H^2 = I$$ and don't really know how and where to use this.

• You are correct that $H^2 = I$. Have you written down the expansion of the matrix exponential and plugged in $H$? What do you get? – NNN Mar 2 '20 at 0:55
• Have you done anything similar in class with, for example $\sigma_x$ or $\sigma_z$? – DaftWullie Mar 2 '20 at 8:03

You are correct that $$H^2 = I$$. Let's set $$a:=\frac{-2it\pi}{h}$$ for simplicity. Then the definition of the matrix exponential gives us
$$U(t) = \sum_{n=0}^\infty \frac{a^n}{n!}H^n$$
Can you use $$H^2 = I$$ to help evaluate this?
The exponential of an operator is defined with respect to its series expansion. The fact that $$H^2=I$$ will simplify this expansion greatly.