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Pauli Matrices

  1. Feb 18, 2013 #1
    Can any one tell me what is eigen value of product of a vector with pauli matrices i.e
    A.σ where A is an arbitrary vector ?
     
  2. jcsd
  3. Feb 18, 2013 #2

    tiny-tim

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    welcome to pf!

    hi rupesh57272! welcome to pf! :smile:

    i don't follow you :redface:

    A.σ is a vector, so how does it have eigenvalues? :confused:
     
  4. Feb 18, 2013 #3

    dextercioby

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    He means a sort of 'scalar' product, which would be (after performing the sum) a 3x3 matrix which can have eigenvalues.

    [tex] \vec{A}\cdot\vec{\sigma} = A_{x}\sigma_x + A_{y}\sigma_y + A_{z}\sigma_z [/tex].
     
  5. Feb 18, 2013 #4

    tiny-tim

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    ohh!!

    then won't they be eigenspinors rather than eigenvectors, in the directions of ±A, and with eigenvalue |A| ?
     
  6. Feb 18, 2013 #5
    Re: welcome to pf!

    Sorry I forgot to mention that it is scalar product of a Vector and Pauli Spin matrices. What is the Eigen Value of it ?
     
  7. Feb 18, 2013 #6
    I think it should be ±|A|
     
  8. Feb 19, 2013 #7

    tiny-tim

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    sorry, yes, ±|A| :smile:

    eg for Sz, or for S-z, the two eigenspinors are the same …

    spinor in the z direction (which we call spin-up, with positive eigenvector, for Sz and spin-down, with negative eigenvector, for S-z)

    spinor in the minus-z direction (which we call spin-down, with negative eigenvector, for Sz and spin-up, with positive eigenvector, for S-z) :wink:
     
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