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Eigenvectors of exponential matrix (pauli matrix)

  1. Jan 18, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the eigenvectors and eigenvalues of exp(iπσx/2) where σx is the x pauli matrix:
    10
    01

    2. Relevant equations
    I know that σxn = σx for odd n
    I also know that σxn is for even n:
    01
    10

    I also know that the exponential of a matrix is defined as Σ(1/n!)xn where the sum runs from n=0 to infinity

    3. The attempt at a solution

    Using knowledge of the matrix exponential, I can say that exp(iπσx/2) = Σ(1/n!)(iπσx/2)n. I can then split this into two sums, one for even n and one for odd n. This allows me to take the power off the matrices for easier summing. It's then that I get stuck. I've attached a file showing the two sums because it's easier and clearer to show you this way.


    I'd really appreciate any help that can be given.
     

    Attached Files:

  2. jcsd
  3. Jan 18, 2015 #2
    I've realised that when written as the two sums, it's clear that there are only two linearly independent eigenvectors for each sum. I'm not sure how it works when the i's get multiplied in though. Aren't there infinite eigenvalues to reflect the infinite nature of the sums?
     
  4. Jan 18, 2015 #3

    BvU

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    Gold Member

    Hi chap, welcome to PF :),

    What would an eigenvector of exp(iπσx/2) look like ? What conditions would it have to satisfy ?

    And what about eigenvectors of σx itself ?
     
  5. Jan 18, 2015 #4
    So the eigenvectors would have to satisfy exp(iπσx/2)v = Av where A is the eigenvalue. I know that eigenvectors of σx are (1,1) and (1,-1)
    Thanks for your help.
     
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