An initial to general state problem (QM Help)

[Brad_Ad23]

1. A 3 level system starts at time t = 0 in the state

$\left|\psi_0\right> = \frac{1}{\sqrt2} \left(\begin{array}{cc}1\\1\\0\end{array}\right)$

The Hamiltonian is $H = 3\left(\begin{array}{ccc}1&0&0\\0&1&1\\0&1&1\end{array}\right)$

If $\hbar = 1$
find the state $\left|\psi_t\right>$ of the system for any time t > 0.

Homework Equations

The Attempt at a Solution

It has been quite a few years since I've done this sort of thing. My approach, perhaps naively, was first to find the eigenvalues for H and then attempt to construct eigenvectors.
I wound up with $E_n = 0, 3, 6$ for eigenvalues and eigenvectors [un-normalized?] of

$\left(\begin{array}{c}0\\1\\-1\end{array}\right) , \left(\begin{array}{c}1\\0\\0\end{array}\right) , \left(\begin{array}{c}0\\1\\1\end{array}\right)$

I am a bit suspicious of the 2nd one, and at any rate I can't seem to remember what happens next (assuming this is the proper approach). I think the eigenvectors should be normalized and then the initial state written as a linear combo of the eigenvectors, with the finale being to add on the basic time-dependence factor of $e^{-i E_n t}$ since h-bar is set to 1 here. Am I on the right approach? If I am or if I am not further help would be appreciated!

 

[Brad_Ad23]

Ok, working on it a bit more I think I figured it out (amazing what taking a break for dinner may do).

So I have the unnormalized eigenvectors. If I normalize them I should wind up with:

$\left|s_1\right> = \frac{1}{\sqrt2}\left(\begin{array}{c}0\\1\\-1\end{array}\right)$
$\left|s_2\right> = \left(\begin{array}{c}1\\0\\0\end{array}\right)$
$\left|s_3\right> = \frac{1}{\sqrt2}\left(\begin{array}{c}0\\1\\1\end{array}\right)$

So $\left|\psi_0\right> = \frac{1}{\sqrt2}\left|s_2\right> + \frac{1}{2}\left(\left|s_1\right> + \left|s_3\right>\right) = \frac{1}{\sqrt2}\left(\begin{array}{c}1\\1\\0\end{array}\right)$

And then to get the general one for some t > 0, I should just put the appropriate exp(iE_nt) in front of each eigenvector yes?

 

[nrqed]

Ok, working on it a bit more I think I figured it out (amazing what taking a break for dinner may do).

So I have the unnormalized eigenvectors. If I normalize them I should wind up with:

$\left|s_1\right> = \frac{1}{\sqrt2}\left(\begin{array}{c}0\\1\\-1\end{array}\right)$
$\left|s_2\right> = \left(\begin{array}{c}1\\0\\0\end{array}\right)$
$\left|s_3\right> = \frac{1}{\sqrt2}\left(\begin{array}{c}0\\1\\1\end{array}\right)$

So $\left|\psi_0\right> = \frac{1}{\sqrt2}\left|s_2\right> + \frac{1}{2}\left(\left|s_1\right> + \left|s_3\right>\right) = \frac{1}{\sqrt2}\left(\begin{array}{c}1\\1\\0\end{array}\right)$

And then to get the general one for some t > 0, I should just put the appropriate exp(iE_nt) in front of each eigenvector yes?

looks perfect to me