Lie Algebra su(2)
Commutation relations [Jᵢ, Jⱼ] = iεᵢⱼₖJₖ — spin representations and the Bloch sphere
Bloch sphere — SU(2) acts on spinors
State |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩
Polar angle θ:
90°
Azimuthal φ:
0°
Rotation axis:
Jₓ
Jᵧ
J_z
Rotation angle α:
0°
Representation j =
1/2
1
3/2
2
Commutation Relations
[J
x
, J
y
] = i J
z
[J
y
, J
z
] = i J
x
[J
z
, J
x
] = i J
y
[J
+
, J
-
] = 2J
z
[J
z
, J
±
] = ±J
±
Casimir: J² = j(j+1)·I