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Consider the following state vector and Hamiltonian:
|\psi (0) \rangle = \frac{1}{5}\left (\begin{array}{cc}3\\0\\4\end{array}\right )
\hat{H} = \left (\begin{array}{ccc}3&0&0\\0&0&5\\0&5&0\end{array}\right )
|\psi (0) \rangle = \frac{1}{5}\left (\begin{array}{cc}3\\0\\4\end{array}\right )
\hat{H} = \left (\begin{array}{ccc}3&0&0\\0&0&5\\0&5&0\end{array}\right )
- If we measure energy, what values can we obtain and with what probabilities?
- Find the state of the system at a later time t; express |\psi (0) \rangle in terms of the eigenvectors of \hat{H}.
- Find the total energy at t = 0 and at a later time t.
- We can get values of either 3 or 5. I don't know how to get the probabilities.
- The state of the system would be:
|\psi _t\rangle = \hat{H}|\psi (0)\rangle = (1/5)(9\ \ 20\ 0)^t
Also:
|\psi (0)\rangle = (3/5)(1\ 0\ 0)^t + (4/5)(0\ 0\ 1)^t
- I can only guess that the energy at t=0 is:
\langle \psi (0)|\hat{H}|\psi (0)\rangle = \langle \psi (0) | \psi _t\rangle
and the energy at a later time t is:
\langle \psi _t |\hat{H}|\psi _t \rangle
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