we can take just the first two terms and substitute them into our equation for , recognizing that is the system Hamiltonian, averaged over the interval , we have: After expanding the products and making the appropriate cancellations, we are left with: giving where is the root mean square deviation of the system Hamiltonian averaged over the interval of interest. The sudden approximation is valid when (the probability of finding the system in a state other than that in which is started approaches zero), thus the validity condition is given by which is a statement of the time-energy form of the Heisenberg uncertainty principle. Diabatic passage In the limit we have infinitely rapid, or diabatic passage: The functional form of the system remains unchanged: This is sometimes referred to as the sudden approximation. The validity of the approximation for a given process can be characterized by the probability that the state of the system remains unchanged: Adiabatic passage In the limit we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions, If the system is initially in an eigenstate of , after a period it will have passed into the corresponding eigenstate of . This is referred to as the adiabatic approximation. The validity of the approximation for a given process can be determined from the probability that the final state of the system is different from the initial state: Calculating adiabatic passage probabilities The Landau–Zener formula In 1932 an analytic solution to the problem of calculating adiabatic transition probabilities was published separately by Lev Landau and Clarence Zener, for the special case of a linearly changing perturbation in which the time-varying component does not couple the relevant states (hence the coupling in the diabatic Hamiltonian matrix is independent of time). The key figure of merit in this approach is the Landau–Zener velocity: where is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and and are the energies of the two diabatic (crossing) states. A large results in a large diabatic transition probability and vice versa. Using the Landau–Zener formula the probability, , of a diabatic transition is given by The numerical approach For a transition involving a nonlinear change in perturbation variable or time-dependent coupling between the diabatic states, the equations of motion for the system dynamics cannot be solved analytically. The diabatic transition