Pauli Matrices under rotation

  • Thread starter shehry1
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  • #1
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Homework Statement


Can anyone tell me why Pauli Matrices remain invariant under a rotation.


Homework 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


The Attempt at a Solution

 

Answers and Replies

  • #2
olgranpappy
Homework Helper
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Homework Statement


Can anyone tell me why Pauli Matrices remain invariant under a rotation.


Homework 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


The Attempt at a Solution

Pauli Matrices are just matrices... they are just arrays of numbers. They don't rotate.
 
  • #3
Avodyne
Science Advisor
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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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