Editing Geometric phase (section)

==Berry phase in quantum mechanics ==
In a quantum system at the ''n''-th [[eigenstate]], an [[Adiabatic theorem|adiabatic]] evolution of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] sees the system remain in the ''n''-th eigenstate of the Hamiltonian, while also obtaining a phase factor.  The phase obtained has a contribution from the state's time evolution and another from the variation of the eigenstate with the changing Hamiltonian. The second term corresponds to the Berry phase, and for non-cyclical variations of the Hamiltonian it can be made to vanish by a different choice of the phase associated with the eigenstates of the Hamiltonian at each point in the evolution.

However, if the variation is cyclical, the Berry phase cannot be cancelled; it is [[invariant (physics)|invariant]] and becomes an observable property of the system. By reviewing the proof of the [[adiabatic theorem]] given by [[Max Born]] and [[Vladimir Fock]], in [[European Physical Journal|Zeitschrift für Physik]] '''51''', 165 (1928), we could characterize the whole change of the adiabatic process into a phase term. Under the adiabatic approximation, the coefficient of the ''n''-th eigenstate under adiabatic process is given by
<math display="block">
C_n(t) = C_n(0) \exp\left[-\int_0^t \langle\psi_n(t')|\dot\psi_n(t')\rangle \,dt'\right] = C_n(0) e^{i\gamma_n(t)},
</math>
where <math>\gamma_n(t)</math> is the Berry's phase with respect to parameter ''t''. Changing the variable ''t'' into generalized parameters, we could rewrite the Berry's phase into
<math display="block">
\gamma_n[C] = i\oint_C \langle n, t| \big(\nabla_R |n, t\rangle\big)\,dR,
</math>
where <math>R</math> parametrizes the cyclic adiabatic process. Note that the normalization of <math>|n, t\rangle</math> implies that the integrand is imaginary, so that <math>\gamma_n[C]</math> is real. It follows a closed path <math>C</math> in the appropriate parameter space. Geometric phase along the closed path <math>C</math> can also be calculated by integrating the [[Berry connection and curvature|Berry curvature]] over surface enclosed by <math>C</math>.