Transition Probabilities at Crossing in the Landau-Zener Problem

We obtain probabilities at a crossing of two linearly time-dependent potentials that are constantly coupled to the other by solving a time-dependent Schrödinger 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 $V_0 \rightarrow {\infty}$ . For a finite coupling the probability branching at the crossing is not even and thus a complete probability transfer at $t \rightarrow {\infty}$ is not achieved in the linear potential crossing problem. The Landau-Zener formula for transition probability at $t \rightarrow {\infty}$ is expressed in terms of the probabilities at the crossing.