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 a set of three 2x2 matrices named after physicist Wolfgang Pauli. They are used in quantum mechanics to represent the spin of particles. Under rotation, these matrices can be used to describe the change in spin orientation of a particle.
Under rotation, Pauli matrices transform according to the rotation matrix. This means that the new matrix is equal to the product of the rotation matrix and the original Pauli matrix.
Pauli matrices are important in quantum mechanics because they represent the fundamental property of spin. Under rotation, they allow us to describe how the spin of a particle changes.
Pauli matrices are closely related to quaternions, which are mathematical objects used to describe rotations in three-dimensional space. The three Pauli matrices can be used to create a set of quaternion units, which can then be used to represent rotations.
Yes, Pauli matrices can be extended to higher dimensions. In 3D space, we use three Pauli matrices, but in higher dimensions, we would use more matrices to represent rotations. However, the same principles for transformation and relation to quaternions still hold.