Rotation and Pauli matrices

In summary, the conversation discusses the derivation of the relation \vec{b} \cdot R_A \vec{a} = \vec{b} \cdot \vec{a} cos( \phi) + \hat{n} \cdot (\vec{a} \times \vec{b})sin(\phi) + 2 (\vec{b} \cdot \hat{n})(\vec{a} \cdot \hat{n}) sin^2(\phi /2) from the given formula (valid for any a), A ( \vec{ \sigma } \cdot \vec{a} ) A^{-1} = \vec{ \sigma } \cdot R_A \vec{a}. The conversation also provides
  • #1
Wiemster
72
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I am given the formula (valid for any a)

[tex]A ( \vec{ \sigma } \cdot \vec{a} ) A^{-1} = \vec{ \sigma } \cdot R_A \vec{a}[/tex]

with [itex]A=exp(i \phi \cdot \vec{\sigma} /2) = exp(i \phi \vec{\sigma} \cdot \hat{n} /2)[/tex] R_A the rotation matrix and sigma the Pauli matrices.

And am supposed to derive the relation

[tex]\vec{b} \cdot R_A \vec{a} = \vec{b} \cdot \vec{a} cos( \phi) + \hat{n} \cdot (\vec{a} \times \vec{b})sin(\phi) + 2 (\vec{b} \cdot \hat{n})(\vec{a} \cdot \hat{n}) sin^2(\phi /2)[/tex]

As a hint some formulas for the traces of products of Pauli matrices are supplied. I have absolutey no idea how to begin and what the problem has to do with the traces of Pauli matrices. Can anybody see where to start?!
 
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  • #2
Hint: Let [itex]\vec{b}[/itex] be an arbitrary vector. Then use the formula for the trace of two Pauli matrices to show that [itex]R_A \vec{a} = \vec{a} cos(\phi) + \hat{n} \times \vec{a} sin(\phi) + 2 (\vec{a} \cdot \hat{n}) \hat{n} sin^2(\phi /2)[/itex]. Multiplying both sides by [itex]\vec{b}[/itex] and using the dot product identity should then give the desired result.
 

1. What is rotation in physics?

Rotation in physics refers to the circular or spinning motion of an object around an axis or center point. It is a fundamental concept in classical mechanics and is used to describe the movement of objects in space.

2. What are Pauli matrices used for?

Pauli matrices are mathematical tools used in quantum mechanics to represent the spin of particles. They are also used in the study of rotation and other physical phenomena, as they can be used to describe the behavior of systems with two possible states.

3. How do you rotate a vector using Pauli matrices?

To rotate a vector using Pauli matrices, you multiply the vector by the appropriate rotation matrix. The rotation matrix is a combination of the Pauli matrices and the rotation angle, and it determines the new direction and magnitude of the rotated vector.

4. What is the relationship between rotation and the Pauli exclusion principle?

The Pauli exclusion principle states that no two identical fermions (particles with half-integer spin) can occupy the same quantum state simultaneously. This principle is related to rotation as it is a result of the symmetries of rotating systems, specifically the rotation of spin in quantum mechanics.

5. Can Pauli matrices be used to describe macroscopic rotations?

No, Pauli matrices are specific to quantum mechanical systems and cannot be used to describe macroscopic rotations. They are only applicable at the atomic and subatomic level, where quantum mechanics principles apply.

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