An initial to general state problem (QM Help)

  • Thread starter Brad_Ad23
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  • #1
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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.



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!
 

Answers and Replies

  • #2
502
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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?
 
  • #3
nrqed
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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?

looks perfect to me:cool:
 

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