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Proof of the Landau-Zener formula. (English) Zbl 0814.35109

Summary: We consider the time dependent Schrödinger 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 nondegenerate eigenvalues which display an avoided crossing of finite minimum gap. We prove that the probability of a quantum transition between these two nondegenerate eigenvalues is given in the adiabatic limit by the well-known Landau-Zener formula.

MSC:

35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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