- #1

Brad_Ad23

- 502

- 1

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

[itex]\left|\psi_0\right> = \frac{1}{\sqrt2} \left(\begin{array}{cc}1\\1\\0\end{array}\right)[/itex]

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

If [itex]\hbar = 1[/itex]

find the state [itex]\left|\psi_t\right>[/itex] of the system for any time t > 0.

[itex]\left|\psi_0\right> = \frac{1}{\sqrt2} \left(\begin{array}{cc}1\\1\\0\end{array}\right)[/itex]

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

If [itex]\hbar = 1[/itex]

find the state [itex]\left|\psi_t\right>[/itex] 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 [itex]E_n = 0, 3, 6[/itex] for eigenvalues and eigenvectors [un-normalized?] of

[itex]

\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)[/itex]

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 [itex]e^{-i E_n t}[/itex] 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!