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!

Eigenvectors of a 2D hermitian operator (general form)

  1. Feb 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Calculate the eigenvectors and eigenvalues of the two-dimensional
    matrix representation of the Hermitean operator [itex]\hat{O}[/itex]
    given by

    |v_k'>[itex]\left(O[/itex]|v_k>= {{O_11,O_12},{O_21,O_22}}


    where all Oij are real. What does Hermiticity imply for the o -
    diagonal elements O12 and O21? For the eigenvectors you may
    assume that the two base vectors read

    v_1= (1) v_2= (0)
    ........(0)...........(1)

    2. Relevant equations
    None


    3. The attempt at a solution

    So, I know how to work out eigenvectors/values:

    |A-λI|=0

    so, I end up with (O11-λ)(O22-λ)-O12O21=0 as the characteristic equation

    with solution λ= 1/2(O22+O11)±1/2√((O22+O11)^2-4(O11O22-O21O12))


    the matrix O, being hermitian, makes O12=O21*

    but how do I progress on to an actual answer?
     
  2. jcsd
  3. Feb 7, 2012 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Since the matrix is real, you know that O21* = O21. You should be able to show that
    $$\lambda = \frac{(O_{11}+O_{22}) \pm \sqrt{(O_{11}-O_{22})^2 + 4O_{12}^2}}{2}$$ That's about as simple as you can get it. Now you want to solve for the eigenvectors. You just have to grind through the algebra.
     
  4. Feb 7, 2012 #3
    But it just ends up as a horrible mess which cannot be the answer!
     
  5. Feb 8, 2012 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    It's not that bad. You need to adjust your threshold for pain. :wink:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Eigenvectors of a 2D hermitian operator (general form)
  1. Hermitian Operators (Replies: 3)

  2. Hermitian Operators (Replies: 13)

  3. Hermitian Operators (Replies: 1)

Loading...