Solving Hamiltonian Problem for 3 State System

[rockstar101]

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.

 

[Feldoh]

You're missing your two other eigenvalues, you need the energies for all 3 states.

 

[rockstar101]

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?

 

[Feldoh]

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.

 

[rockstar101]

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.

 

[rockstar101]

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?