Why are Pauli Matrices Invariant under Rotation?

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SUMMARY

Pauli Matrices remain invariant under rotation due to the specific transformation properties of spinors and the corresponding rotation operator. The rotation operator can be expressed as U = I*Cos(x/2) - i*(pauli matrix).(unit vector)*Sin(x/2), where x represents the angle of rotation. The relationship sigma_i = R_ij (U sigma_j U^dagger) illustrates how both the spinor indices and the vector index must be rotated simultaneously to maintain invariance. This conclusion is supported by the reference to Sakurai's Quantum Mechanics text, specifically equation 3.2.44.

PREREQUISITES
  • Understanding of quantum mechanics, specifically spinors and their transformation properties.
  • Familiarity with Pauli Matrices and their mathematical representation.
  • Knowledge of rotation operators in quantum mechanics.
  • Basic grasp of matrix operations and exponentiation in the context of linear algebra.
NEXT STEPS
  • Study the derivation of the rotation operator for spin systems in quantum mechanics.
  • Explore the implications of Pauli Matrices in quantum state transformations.
  • Learn about the mathematical foundations of spinors and their applications in quantum theory.
  • Review Sakurai's Quantum Mechanics, focusing on sections related to angular momentum and rotation operators.
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Students and professionals in quantum mechanics, physicists studying spin systems, and anyone interested in the mathematical foundations of quantum rotations.

shehry1
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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

 
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shehry1 said:

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.
 
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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