1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding eigenstates and eigenvalues of hamiltonian

  1. Mar 12, 2013 #1
    Hey there, the question i'm working on is written below:-

    Let |a'> and |a''> be eigenstates of a Hermitian operator A with eigenvalues a' and a'' respectively. (a'≠a'') The Hamiltonian operator is given by:

    H = |a'>∂<a''| + |a''>∂<a'|

    where ∂ is just a real number.

    Write down the eigenstates of the hamiltonian. what are their energy eigenvalues?

    Was feeling a bit confused by the question at first, and was just wondering if someone could let me know if my thoughts so far are on the right track?

    Essentially, I just wrote the eigenstates as the kets: |E[itex]_{a'}[/itex]> and |E[itex]_{a''}[/itex]>. In order to find the eigenvalues of the energy, I constructed the matrix:

    H = [tex]
    0 & ∂\\
    ∂ & 0

    so that I could use the general det(H - λI) = 0 to find the eigenvalues. is this reasoning vaguely in the correct direction?

    Also, was just wondering, if in the original hamiltonian equation, i'm allowed to take the ∂ symbol out (written below) because it is just a real number?

    H = ∂(|a'><a''| + |a''><a'|)
  2. jcsd
  3. Mar 12, 2013 #2
    Yes to both, you are headed in the right direction
  4. Mar 14, 2013 #3
    thanks for that. i have actually continued on with this problem, and i've come across another question.

    i found the eigenvalues to be ±∂. the problem then asks

    b) suppose the system is known to be in state |a'> at t=0. write down the schrodinger picture for t>0.

    Which my working out is:|α, t[itex]_{0}[/itex]=0> = |a'>

    |α, t[itex]_{0}[/itex]=0; t> = U(t,t[itex]_{0}[/itex])|a'>
    =exp( (-iE[itex]_{a'}[/itex]t) / h-bar )|a'>
    =exp( -i∂t / h-bar )|a'>

    question c)
    what is the probability for finding the system in state |a''> for t>0, if the system is known to be in state |a'> at t=0?

    for this, i was thinking that for |a''>= U(t,t[itex]_{0}[/itex])|a'> = =exp( -i∂t / h-bar )|a'>

    and for the probability, i need to calculate |<a''|a'>|[itex]^{2}[/itex] = |(exp( -i∂t / h-bar ) <a'| )(|a'>)|[itex]^{2}[/itex], where <a'|a'> = 1

    am i allowed to do this??? i'm probably ignoring something important....
  5. Mar 15, 2013 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The state ##\lvert a'\rangle## isn't an eigenstate of the Hamiltonian, so it's not correct to say it has an energy ##E_{a'}##. You need to express ##\lvert a'\rangle## in terms of the eigenstates of H.
  6. Mar 15, 2013 #5
    oh ok then. would this be the right plan of action then?

    using the eigenvectors i found for the hamiltonian|E[itex]_{1}[/itex]> = (1,1) and |E[itex]_{2}[/itex]> = (1,-1). i then constructed:

    ||E[itex]_{1}[/itex]> = |a'> + |a''> and
    |E[itex]_{2}[/itex]> = |a'> - |a''> ( i have left out the normalization constant here)

    then i can have |a'> and |a''> written in terms of the eigenstates of H. back in the schrodinger picture, can i then substitute |a'> for |E[itex]_{1}[/itex]>+|E[itex]_{2}[/itex]>, and then use E[itex]_{1}[/itex] & E[itex]_{1}[/itex] in the exponential? ==>

    |α, t[itex]_{0}[/itex]=0; t> = U(t,t[itex]_{0}[/itex])|a'>
    =exp((-iE[itex]_{a}[/itex]′t) / h-bar)|a'>
    =(exp(-i∂t/h-bar)||E[itex]_{1}[/itex]> + exp(i∂t/h-bar)||E[itex]_{2}[/itex]>

    then sub |a'> and |a''> back into this equation?
  7. Mar 15, 2013 #6


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Yes, that's exactly what you want to do.
  8. Mar 15, 2013 #7
    yay! thanks for your help :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted