probability can still be obtained using one of the wide variety of numerical solution algorithms for ordinary differential equations. The equations to be solved can be obtained from the time-dependent Schrödinger equation: where is a vector containing the adiabatic state amplitudes, is the time-dependent adiabatic Hamiltonian, and the overdot represents a time derivative. Comparison of the initial conditions used with the values of the state amplitudes following the transition can yield the diabatic transition probability. In particular, for a two-state system: for a system that began with . See also Landau–Zener formula Berry phase Quantum stirring, ratchets, and pumping Adiabatic quantum motor Born–Oppenheimer approximation Diabatic Eigenstate thermalization hypothesis References Theorems in quantum mechanics