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

Question

Consider a Hamiltonian $H = sigma_x otimes sigma_z$
Construct the time evolution operator $U(t) = mathrm{e}^{-frac{iHt}{frac{h}{2pi}}}$ [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.

Paul Nation · Answer

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

Alex · Answer

Here is a hint:

You are correct that $H^2 = I$. Let's set $a:=frac{-2itpi}{h}$ 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?

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

2 Answers

Add your own answers!

Ask a Question