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Eigenvalues and eigenvectors, pauli matrices

  1. May 26, 2015 #1
    1. The problem statement, all variables and given/known data

    Look at the matrix:

    A = sin t sin p s_x + sin t sin p s_y +cos t s_z

    where s_i are the pauli matrices

    a) Find the eigenvalues and normalized eigenvectors (are they orthogonal)?

    b) Write the eigenvector of s_x with positive eigenvalue as a linear combination of the eigenvalues of A

    2. Relevant equations


    3. The attempt at a solution

    I got an answer but I am not confident as to it's veracity.

    The matrix A is :

    (cos t , sin t cos p - i sin t sin p
    sin t cos p + i sin t sin p, - cos t)

    or

    (cos t, sin t (e^-i p)
    sin t (e^ip), - cos t)

    Finding the eigenvalues

    (cos t - lambda, sin t (e^i p)
    sin t (e^i p), - cos t - lambda)

    You get

    lambda^2 - 1 = 0

    So the eigenvalues are +-1

    The problem is finding the eigenvectors;

    For +1

    (cos t - 1, sin t e^(-i p )
    sin t e^i p, - cos t - 1)

    Using the cofactor method you get the equations

    (cos t - 1) x +(sin e^i p) y = 0

    (sin t e^i p) x + (-cos t - 1) y = 0

    I get

    v1 = C_1 (-e^-i p tan (t/2), 1)
    v2 = C_2 (-e^i p tan (t/2), 1)

    Then normalizing

    C_1 = 1/sqrt(e^-2 i p tan (t/2)^2 + 1)

    C_2 = 1/sqrt(e^-2 i p cot (t/2)^2 + 1)

    I don't know how to do b

    Any help/checking would be helpful
     
  2. jcsd
  3. May 26, 2015 #2

    vela

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    Staff Emeritus
    Science Advisor
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    Looks good so far. Try using the trig identities ##\sin^2 \frac \theta 2 = \frac{1 - \cos \theta}{2}## and ##\sin \theta = 2 \sin\frac \theta 2 \cos \frac \theta 2##. That'll make the algebra a bit easier to deal with.
     
  4. May 28, 2015 #3
    Thanks for all your help guys!
     
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