1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
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)

  1. Aug 7, 2008 #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.

    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

    \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!
  2. jcsd
  3. Aug 7, 2008 #2
    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)[/itex]
    \left|s_2\right> = \left(\begin{array}{c}1\\0\\0\end{array}\right)[/itex]
    \left|s_3\right> = \frac{1}{\sqrt2}\left(\begin{array}{c}0\\1\\1\end{array}\right)[/itex]

    So [itex]\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(iEnt) in front of each eigenvector yes?
  4. Aug 8, 2008 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    looks perfect to me:cool:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook