I'm learning about Berry phase for a presentation and am working through Berry's original paper.

I can't quite make the connection between equation 4

$$ \dot{\gamma}(t) = i \langle n(\vec{R}) |\nabla_R |n(\vec{R})\rangle \cdot \dot{\vec{R}}(t) $$

where and equation 6

$$ {\gamma}(C) = i \oint_C \langle n(\vec{R}) |\nabla_R |n(\vec{R})\rangle \cdot d{\vec{R}}(t) $$

Equation 4 is derived from postulating that another phase (other than from Hamiltonian evolution) exists, and if so then that phase will obey that form. He then essentially jumps to equation 6.

I assume there is some calculus relation or detail from the adiabatic approximation that I am not seeing, any ideas?

I'm learning about Berry phase for a presentation and am working through Berry's original paper.

I can't quite make the connection between equation 4

$$ \dot{\gamma}(t) = i \langle n(\vec{R}) |\nabla_R |n(\vec{R})\rangle \cdot \dot{\vec{R}}(t) $$

where and equation 6

$$ {\gamma}(C) = i \oint_C \langle n(\vec{R}) |\nabla_R |n(\vec{R})\rangle \cdot d{\vec{R}} $$

Equation 4 is derived from postulating that another phase (other than from Hamiltonian evolution) exists, and if so then that phase will obey that form. He then essentially jumps to equation 6.

I assume there is some calculus relation or detail from the adiabatic approximation that I am not seeing, any ideas?