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Pauli Matrices under rotation

  1. Nov 22, 2008 #1
    1. The problem statement, all variables and given/known data
    Can anyone tell me why Pauli Matrices remain invariant under a rotation.

    2. Relevant equations
    Probably the rotation operator in the form of the exponential of a pauli matrix having an arbitrary unit vector as its input. It may also be written as:
    I*Cos(x/2) - i* (pauli matrix).(unit vector) * Sin(x/2) where x is the angle of rotation.

    See Sakurai 3.2.44

    3. The attempt at a solution
  2. jcsd
  3. Nov 22, 2008 #2


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    Homework Helper

    Pauli Matrices are just matrices... they are just arrays of numbers. They don't rotate.
  4. Nov 22, 2008 #3


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    Science Advisor

    It's because you need to rotate both the spinor indices AND the vector index; let
    U = I*Cos(x/2) - i* (pauli matrix).(unit vector) * Sin(x/2) where x is the angle of rotation, and let R_ij be the corresponding matrix that would rotate a vector by the angle x about the unit vector. Then

    sigma_i = R_ij (U sigma_j U^dagger)

    where j is summed and the spinor indices are suppressed.
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