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Communications in Mathematical Physics

, Volume 136, Issue 3, pp 433–449 | Cite as

Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps

  • George A. Hagedorn
Article

Abstract

We consider a smooth operator-valued functionH(t,δ) that has two isolated non-degenerate eigenvaluesE A (t,δ) andE B (t,δ) for δ>0. We assume these eigenvalues are bounded away from the rest of the spectrum ofH(t,δ), but have an avoided crossing with one another with a closest approach that isO(δ) as δ tends to zero. Under these circumstances, we study the small ε limit for the adiabatic Schrödinger equation
$$i\varepsilon \frac{{\partial \psi }}{{\partial t}} = H(t,\varepsilon ^{1/2} )\psi .$$
We prove that the Landau-Zener formula correctly describes the coupling between the adiabatic states associated with the eigenvaluesE A (t,δ) andE B (t,δ) as the system propagates through the avoided crossing.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • George A. Hagedorn
    • 1
    • 2
  1. 1.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Center for Transport Theory and Mathematical PhysicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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