rotation matrices
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Short Word note by Phil dated 1.23.08 collecting rotation matrix formulas. It covers the SU(2) two-dimensional case with generators J = σ/2 and R = exp(-iθ n·J) = cos(θ/2) - i σ·n sin(θ/2), then the SO(3) Cartesian case with generators (Ji)jk = -i εijk, the general axis-angle rotation matrix, a trace formula for Rij, and the relation to the Wigner D(1) matrices through a unitary change of basis. Many matrices were lost in text extraction.
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Rotation Matrices PhL 1.23.08
1. Two dimensions as in SU(2):
generators: Ji = (1/2)i rotations: R() = exp(-iJ)
Jx = (1/2) Jy = (1/2) Jz = (1/2)
Rx = Ry = Rz =
R() = exp(-iJ) = exp(-i/2σ) = cos(θ/2) – i σ sin(θ/2) since (σ)2 = 1
2. Three dimensions Cartesian as in SO(3):
Jx = Jy = Jz =
Rx() = Ry() = Rz() =
(Ji)jk = -i ijk. [Ji, Vj ] = iijkVk R V R-1 = R-1 V
Special case of the above: R J R-1 = R-1 J for any J representation
[R()]kl = cos kl + ksl ns sin + nk nl (1 - cos)
Rij = (1/2) tr [ iRjR†]
Ry() = U d(1)() U-1
Rz() Ry() Rz() = U D(1)(,,) U-1 U = <k|1,m> = Ukm
Transcription of the 3D rotation matrices:
Rx(θ) = Ry(θ) = Rz(θ) =
Here is a 2D Rz(θ) matrix
Rz(θ) =