Transition Probabilities at Crossing in the Landau-Zener Problem

摘要

We obtain probabilities at a crossing of two linearly time-dependent potentials that are constantly coupled to the other by solving a time-dependent Schrodinger equation.We find that the system which was initially localized at one state evolves to split into both states at the crossing.The probability splitting depends on the coupling strength V_0 such that the system stays at the initial state in its entirety when V_0 = 0 while it is divided equally in both states when V_0 -> infinity.For a finite coupling the probability branching at the crossing is not even and thus a complete probability transfer at t -> infinity is not achieved in the linear potential crossing problem.The Landau-Zener formula for transition probability at t->infinity is expressed in terms of the probabilities at the crossing.