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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

    [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!
     
  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:

    [itex]
    \left|s_1\right> = \frac{1}{\sqrt2}\left(\begin{array}{c}0\\1\\-1\end{array}\right)[/itex]
    [itex]
    \left|s_2\right> = \left(\begin{array}{c}1\\0\\0\end{array}\right)[/itex]
    [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)
    [/itex]

    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

    nrqed

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    looks perfect to me:cool:
     
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