1. Not finding help here? Sign up for a free 30min 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!

Exercise with Hamiltonian matrix

  1. Sep 1, 2016 #1
    1. The problem statement, all variables and given/known data

    I have the matrix form of the Hamiltonian:

    H = ( 1 2-i
    2+i 3)

    If in the t=0, system is in the state a = (1 0)T, what is Ψ(x,t)?
    2. Relevant equations

    Eigenvalue equation

    3. The attempt at a solution

    So, I have diagonalized given matrix and got the eigenvalues: 2+√6 and 2-√6. I am suspecting that these are not good, since I can't get eigenvectors I can use. When trying to calculate eigenvectors, I get:
    a = 1 and b=(-1+√6)/(2+i). This is the one I got when I used 2+√6 but after that I didn't even try with the other eigenvalue since it will be similar.
    I don't know what to do with those and don't know how to normalize them. Also, even if I knew how to get correct eigenvectors, I am not sure how to proceed and get Ψ(x,t).
     
  2. jcsd
  3. Sep 1, 2016 #2

    DrClaude

    User Avatar

    Staff: Mentor

    I don't see how you got that value for b (I don't get the same signs), so check your math. If what bothers you is the (2+i) in the denominator, simply multiply by (2-i)/(2-i).
     
  4. Sep 1, 2016 #3
    Do you get the same eigenvalues? I put it in the form:
    ( 1 2-i
    2+i 3 ) * (a b)t = (2+√6) (a b)T

    I'm sorry, I don't know to write it properly.

    Then I got:
    a+(2-i)b = (2+√6)a
    That's where the expression comes from. And by using the second equation i got a=a in which cases we always put 1 in our class. Is that wrong?
     
  5. Sep 1, 2016 #4

    DrClaude

    User Avatar

    Staff: Mentor

    That gives
    $$
    b = \frac{1 + \sqrt{6}}{2 - i}
    $$
    which is not what you wrote above.

    Try multiplying with ##(2 + i)/(2 + i)##.
     
  6. Sep 2, 2016 #5
    Ok, I got the vectors but don't know how to proceed to get Ψ (x,t)
     
  7. Sep 2, 2016 #6

    DrClaude

    User Avatar

    Staff: Mentor

    What have you learned about time evolution?
     
  8. Sep 2, 2016 #7
    To be precise, I don't understand how to get right coefficients to write Ψ(x,0) as a linear combination of the vectors I got. When i get that, I believe that I just have to add time dependence e-iEt/†
     
  9. Sep 2, 2016 #8

    DrClaude

    User Avatar

    Staff: Mentor

    Scalar product. You have found the eigenvectors φ1 and φ2 and want to write Ψ(t=0) = c1 φ1 + c2 φ2, you find the coefficients using
    $$
    c_n = \phi_n^\dagger \Psi
    $$
     
  10. Sep 4, 2016 #9
    Is that equal to finding the norm of the eigenvectors? For example, I have three eigenvectors: v1 = v2 = (1 0 0) and v3=1/√2 (0 -i 1).

    Would I write Ψo= 2*(1 0 0)+1/√2 (0 -i 1)?
     
  11. Sep 4, 2016 #10
    I have used different example, since I get too complicated vectors in the exercise I posted originally
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Exercise with Hamiltonian matrix
  1. Matrix Hamiltonian? (Replies: 1)

  2. Hamiltonian Matrix? (Replies: 4)

Loading...