(adsbygoogle = window.adsbygoogle || []).push({}); 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.

2. Relevant equations

3. 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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: An initial to general state problem (QM Help)

**Physics Forums | Science Articles, Homework Help, Discussion**