The Quantum Adiabatic Approximation and the Geometric Phase

A precise definition of an adiabaticity parameter ν of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(τ) = ∑ l U (l)(τ) with U (l)(τ) being at least of the order ν. In particular U (0)(τ) corresponds to the adiabatic approximation and yields Berry’s adiabatic phase. It is shown that this series expansion has nothing to do with the 1/τ -expansion of U(τ). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. Some related issues concerning the geometric phase are also discussed. E-mail: [email protected] 1