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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.

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  • $\begingroup$ 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? $\endgroup$ – Alex Mar 2 at 0:55
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    $\begingroup$ Have you done anything similar in class with, for example $\sigma_x$ or $\sigma_z$? $\endgroup$ – DaftWullie Mar 2 at 8:03
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Here is a hint:

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?

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The exponential of an operator is defined with respect to its series expansion. The fact that $H^2=I$ will simplify this expansion greatly.

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