Proof of the Landau–Zener formula

Abstract

We consider the time dependent Schrodinger equation in the adiabatic limit when the Hamiltonian is an analytic unbounded operator. It is assumed that the Hamiltonian possesses for any time two instantaneous non-degenerate eigenvalues which display an avoided crossing of finite minimum gap. We prove that the probability of a quantum transition between these two non-degenerate eigenvalues is given in the adiabatic limit by the well-known Landau–Zener formula.