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)=${\sum }_{\mathrm{\ell }}$ ${\mathrm{U}}^{\left(\mathrm{\ell }\right)}$(τ) with ${\mathrm{U}}^{\left(\mathrm{\ell }\right)}$(τ) being at least of the order ${\nu }^{\mathrm{\ell }}$. In particular, ${\mathrm{U}}^{\mathrm{}\left(0\right)\mathrm{}}$(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. This suggests the introduction of an adiabatic product expansion for U(τ) which turns out to yield exact expressions for U(τ) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schrödinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.