Invariance of Pauli-matrices under rotation

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  • #1
NewGuy
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I'm trying to prove that the helicity operator [itex]\pmb{\sigma}\cdot\pmb{\hat{p}}[/itex] is invariant under rotations. I found in Sakurai: Modern Quantum Mechanics page 166 that the Pauli matrices are invariant under rotations. Clearly that is sufficient for the helicity operator to be invariant under rotations. However I'm unable to prove that the Pauli matrices are invariant under rotations, and Sakurai states no proof. How would I prove this? I do know the fact that if U is the unitary matrix that represents rotation from n to n', then [itex]U(\pmb{\sigma}\cdot\pmb{n})U^\dagger=\pmb{\sigma}\cdot\pmb{n'}[/itex], however it doesn't seem to help.
 

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  • #2
xlines
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I'm trying to prove that the helicity operator [itex]\pmb{\sigma}\cdot\pmb{\hat{p}}[/itex] is invariant under rotations. I found in Sakurai: Modern Quantum Mechanics page 166 that the Pauli matrices are invariant under rotations. Clearly that is sufficient for the helicity operator to be invariant under rotations. However I'm unable to prove that the Pauli matrices are invariant under rotations, and Sakurai states no proof. How would I prove this? I do know the fact that if U is the unitary matrix that represents rotation from n to n', then [itex]U(\pmb{\sigma}\cdot\pmb{n})U^\dagger=\pmb{\sigma}\cdot\pmb{n'}[/itex], however it doesn't seem to help.

I seem to remember of proving something similar. I'll dig up my QM notes and try to clear thing up, unless someone answers by the time I get to my office.
 
  • #3
NewGuy
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If you would that I would be very grateful :)
 
  • #4
xlines
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I'm trying to prove that the helicity operator [itex]\pmb{\sigma}\cdot\pmb{\hat{p}}[/itex] is invariant under rotations. I found in Sakurai: Modern Quantum Mechanics page 166 that the Pauli matrices are invariant under rotations. Clearly that is sufficient for the helicity operator to be invariant under rotations. However I'm unable to prove that the Pauli matrices are invariant under rotations, and Sakurai states no proof. How would I prove this? I do know the fact that if U is the unitary matrix that represents rotation from n to n', then [itex]U(\pmb{\sigma}\cdot\pmb{n})U^\dagger=\pmb{\sigma}\cdot\pmb{n'}[/itex], however it doesn't seem to help.

I am sorry, but I will fail you too. What I did is to solve problem 1.3. from Sakurai where it is required to show that determinant of [itex]\pmb{\sigma}\cdot\pmb{n}[/itex] is invariant under operation you quoted. I used 3.2.34, 35, 39 and 44.

Middle result of this solution that may help you is:

[itex]U(\pmb{\sigma}\cdot\vec{a})U^\dagger=\pmb{\sigma}\cdot (\vec{a} cos \phi + 2 \hat{n} (\hat{n} \vec{a}) sin^{2}(\phi /2) - (\hat{n} \times\vec{a}) sin \phi ) [/itex]

Where U is given by 3.2.44. Hope it helps to any amount, I wish you luck with your problem.
 

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