# Obtain Time evolution from Hamiltonian

## Homework Statement

A quantum system with a ##C^3## state space and a orthonormal base ##\{|1\rangle, |2\rangle, |3\rangle\}## over which the Hamiltonian operator acts as follows:
##H|1\rangle = E_0|1\rangle+A|3\rangle##
##H|2\rangle = E_1|2\rangle##
##H|3\rangle = E_0|3\rangle+A|1\rangle##

Build H and obtain the eigenvalues and the eigenvectors.

If ##|\Phi((0)\rangle = |2\rangle## obtain ##|\Phi(t)\rangle##.

If ##|\Phi((0)\rangle = |3\rangle## obtain ##|\Phi(t)\rangle##.

## The Attempt at a Solution

I've managed to build the H matrix and obtain the eigenvectors.
The second part is ##|\Phi(t)\rangle = e^{-\frac{i}{\hbar}E_1t}|2\rangle##
It is the third part that I'm not sure. Should I build the change of basis matrix from the eigenvectors and apply it to the vector ##|3\rangle## expressed as ## |0 0 1\rangle##, so that I get its eigenvectors descomposition?

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mfb
Mentor
It is the third part that I'm not sure. Should I build the change of basis matrix from the eigenvectors and apply it to the vector ##|3\rangle## expressed as ## |0 0 1\rangle##, so that I get its eigenvectors descomposition?
That should work, yes. Once you have the composition in eigenvectors, the time-evolution is easy to write down.

Thanks mfb.

Just a little side question: I've done the change of basis with ##|3\rangle' = S|3\rangle S^{-1}##. Could I have done ##|3\rangle' = S^\dagger|3\rangle S##?

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