Berry Phase & Geometric Phase

A quantum spin traces a closed loop on the Bloch sphere — acquiring a phase equal to half the solid angle enclosed.

Loop Parameters

γ = 0.00
Berry Phase (radians)
Solid angle Ω = 0.00 sr
γ = Ω/2 = 0.00
Berry Phase: γ = i∮⟨n(R)|∇_R|n(R)⟩·dR

When a quantum state evolves adiabatically around a closed loop in parameter space, it acquires a geometric phase equal to half the solid angle subtended by the loop on the Bloch sphere. This is purely geometric — independent of speed.