In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian H ( R ) depends on a (vector) parameter R that varies with time t . If the n 'th eigenvalue ε ε n ( R ) remains non-degenerate everywhere along the path and the variation with time t is sufficiently slow, then a system initially in the normalized eigenstate | n ( R ( 0 ) ) ⟩ ⟩ will remain in an instantaneous eigenstate | n ( R ( t ) ) ⟩ ⟩ of the Hamiltonian H ( R ( t ) ) , up to a phase, throughout the process. Regarding the phase, the state at time t can be written as | Ψ Ψ n ( t ) ⟩ ⟩ = e i γ γ n ( t ) e − − i ℏ ℏ ∫ ∫ 0 t d t ′ ε ε n ( R ( t ′ ) ) | n ( R ( t ) ) ⟩ ⟩ , where the second exponential term is the "dynamic phase factor." The first exponential term is the geometric term, with γ γ n being the Berry phase. From the requirement that the state | Ψ Ψ n ( t ) ⟩ ⟩ satisfies the time-dependent Schrödinger equation, it can be shown that γ γ n ( t ) = i ∫ ∫ 0 t d t ′ ⟨ ⟨ n ( R ( t ′ ) ) | d d t ′ | n ( R ( t ′ ) ) ⟩ ⟩ = i ∫ ∫ R ( 0 ) R ( t ) d R ⟨ ⟨ n ( R ) | ∇ ∇ R | n ( R ) ⟩ ⟩ , indicating that the Berry phase only depends on the path in the parameter space, not on the rate at which the path is traversed.