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- How do we evaluate the exponential of a hamiltonian involving tensor products? Why do i get a zero result?

**EDIT: I'M SO DUMB! I can't believe I can't multiply matrices together. Of course the result is not zero, the matrix on the left will be:**

$$

\begin{pmatrix}

0 & e^{i\omega_at/2}\\

e^{-i\omega_at/2}&0

\end{pmatrix}

$$

So i was solving problem 3 from https://ocw.mit.edu/courses/physics...iii-spring-2018/assignments/MIT8_06S18ps5.pdf

Here is my working (i used properties 3 and 8 from https://en.wikipedia.org/wiki/Kronecker_product to simplify the exponential of a tensor product):

Clearly 0 does not make sense, but I don't understand where I could have made a mistake? It seems like from the way the perturbation is defined as having off-diagonal elements, there is no way multiplying it by the exponential of the unperturbed hamiltonian, which is diagonal, will yield a non-zero result.

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