An upper bound for the generalized adiabatic approximation error with a superposition initial state

The classical adiabatic approximation theory gives an adiabatic approximate solution to the Schr6dinger equation （SE） by choosing a single eigenstate of the Hamiltonian as the initial state. The superposition principle of quantum states enables us to mathematically discuss the exact solution to the SE starting from a superposition of two different eigenstates of the time-dependent Hamiltonian H（0）. Also, we can construct an approximate solution to the SE in terms of the corresponding instantaneous eigenstates of H（t）. On the other hand, any physical experiment may bring errors so that the initial state （input state） may be a superposition of different eigenstates, not just at the desired eigenstate. In this paper, we consider the generalized adiabatic evolution of a quantum system starting from a superposition of two different eigenstates of the Hamiltonian at t = 0. A generalized adiabatic approximate solution （GAAS） is constructed and an upper bound for the generalized adiabatic approximation error is given. As an application, the fidelity of the exact solution and the GAAS is estimated.