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Calculation with Pauli matrices

  1. Sep 2, 2016 #1
    1. The problem statement, all variables and given/known data

    Hey :-)
    I just need some help for a short calculation.
    I have to show, that
    [tex] (\sigma \cdot a)(\sigma \cdot b) = (a \cdot b) + i \sigma \cdot (a \times b) [/tex]



    3. The attempt at a solution

    I am quiet sure, that my mistake is on the right side, so I will show you my calculation for this one:
    [tex] a_xb_x + a_yb_y+a_zb_z + i\sigma_x (a_yb_z - a_3b_2) + i\sigma_y (a_zb_x-a_xb_z) + i\sigma_z (a_xb_y -a_yb_x) [/tex]

    The last 3 terms are a 2x2 matrix and the first 3 terms are just a scalar...
    So i can`t add them.

    would be happy fora small hint what is wrong :-)
    Thank you
     
  2. jcsd
  3. Sep 2, 2016 #2

    blue_leaf77

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    There is actually an identity matrix to be multiplied wit ##a\cdot b##.
     
  4. Sep 2, 2016 #3
    hey. thank you for your answer.
    Yes, right. that brings to to the result I want.
    Is there a rule, why I have to multiply the result of the dot product with the idendity matrix?
    Because the other terms include a Pauli Matrix and the result
    of the dot produkt must adapt to that structure?
     
  5. Sep 2, 2016 #4

    blue_leaf77

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    Of course it can be proven using the more fundamental properties of Pauli matrices, especially their commutation and anti-commutation. An easy prove can be found here.
     
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