Obtain Time evolution from Hamiltonian

[carllacan]

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 $|\Phi(t)\rangle$.

Homework Equations

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?

 

[mfb]

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.

 

[carllacan]

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 $|3\rangle' = S^\dagger|3\rangle S$?