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Hamiltonian problem

  1. Oct 17, 2009 #1
    1. The problem statement, all variables and given/known data
    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)> ??




    2. Relevant equations

    3. 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.
     
  2. jcsd
  3. Oct 17, 2009 #2
    You're missing your two other eigenvalues, you need the energies for all 3 states.
     
  4. Oct 17, 2009 #3
    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?
     
  5. Oct 17, 2009 #4
    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.
     
  6. Oct 17, 2009 #5
    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.
     
  7. Oct 18, 2009 #6
    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 plz?
     
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