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Eigenvalues, eigenvectors, and time evolution

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A. Given matrix A = 1 2 0 where i found the eigenvalues to be 3hw, 0, -3hw.
2 0 2
0 2 -2

then if at t=0 state-vector is 1 THEN compute the state vector at time t.

Please explain me how to get this, Im very confused. I know its some sort of matrix multiplication but why?

b. If A = 1 0 0 is an observable. FIND its expectation at time t and compute the
0 2 0
0 0 0

probability that measuring A at time t gives the result 1.

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In problem a) the matrix A is presumably the Hamiltonian up to some factor. First you must compute the eigenvalues. You either did that incorrectly or you wrote down A incorrectly. You can see that at once without doing any computations as follows. The trace of a matrix (this is the sum of the elements on the diagonal) is invariant under transformations of the basis vectors. This means that the trace must equal the sum of the eigenvalues. The trace of A is -1, but your set of eigenvalues sums to 0.

Suppose you find the correct eigenvalues and ...

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