Solving Hamiltonian Problem for 3 State System

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Homework Statement


Let
( Eo 0 A )
( 0 E1 0 )
( A 0 Eo )

be the matrix representation of the Hamiltonian for a three state system with basis states
|1> |2> and |3> .
If |ψ(0)> = |3> what is |ψ(t)> ??




Homework Equations



The Attempt at a Solution



First I need to find the energy eigenstate of the system:

H|ψ> = E|ψ> and
(Eo 0 A , 0 E1 0, A 0 Eo)T ( <1|ψ> , <2|ψ>, <3|ψ>)T = E( <1|ψ> , <2|ψ>, <3|ψ>)T

so I got the equation (Eo - E)(E1 - E)(Eo-E) + A^2(E1-E) = 0
simplify, (Eo - E)^2 (E1 - E) + A^2(E1 - E) = 0

for this equation to be true, then E1 = E ... is this my eigenvalue??

From here, how do I find the energy eigenstate?
After that, what should I do to answer the question?
I would really appreciate any hint or help... thank you.
 
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You're missing your two other eigenvalues, you need the energies for all 3 states.
 
Thanks Feldoh, I think my second and third eigenvalues are E = Eo + A and E = Eo - A.
So I'm assuming there will be three eigenstates...
but I really don't have a clue how I can obtain those eigenstates. How can I find the eigenstates?
 
I'm not sure if those are the right eigenvalues, however once you do find the right eigenvalues you'd just solve for states of the Hamiltonian just like any other eigenvector problem.
 
From (Eo - E)^2 (E1 - E) + A^2(E1 - E) = 0

I simplified to get (E1 - E)[ (Eo - E)^2 + A^2] = 0

Thus, Eo - E = +/- A

Hence my three eigenvalues are E = Eo - A, E= Eo + A, E = E1

but I'm having trouble finding the eigenstate because Eo and E1 are different.
 
ok so I figured out the three eigenstates:
for E= E1
eigenstate is |I> = 0|1> + 1|2> + 0 = |2> since the eigenvector is (0 1 0)T

for E= Eo + A
eigenstate is |II> = 1/√2 |1> + 1/√2|3> b/c eigenvector is 1/√2( 1 0 1)T

for E= Eo - A
eigenstate is |III> = 1/√2|1> - 1/√2|3>

now... If |ψ(0)> = |3> what is |ψ(t)> ??

any hint please?
 
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