Landau-Zener transition

From Wikipedia, the free encyclopedia

You have new messages (last change).

The phrase "Landau-Zener transition" refers to the following general problem in quantum dynamics, which was first solved by Landau and Zener in 1932.

We are given a time-dependent Hamiltonian containing only two levels, whose energy difference changes linearly in time, according to \delta E= H_{22}-H_{11}={\dot \epsilon} t. Furthermore, there is a finite transition amplitude H12 = H21 = v for transitions between these levels (which remains constant throughout the dynamics). If we started out in (say) the lower energy eigenstate in the infinite past, what is the probability to find the system in the upper energy eigenstate in the infinite future (i.e. for the system to undergo a so-called Landau-Zener transition)? For infinitely slow variation of the energy difference, no such transition will take place, as the system will always stay in the same instantaneous eigenstate (the one corrresponding to the Hamiltonian at the present moment of time). In general, the following formula describes the probability for the transition:

P_{LZ} = \exp\left[ - 2 \pi {v^2 \over \hbar {\dot \epsilon} } \right]

Note that this expression is not analytic in the limit of vanishing rate of energy change, i.e. a Taylor expansion (perturbation series) in {\dot \epsilon} cannot yield the correct result.

Retrieved from "http://en.wikipedia.org../../../l/a/n/Landau-Zener_transition_a82a.html"