The Riccati equation bridges the two probability amplitudes in Landau-Zener transitions and pinpoints why the Markov approximation succeeds for one amplitude but fails for the other.
(62), to establish the identity |I(τ)| = p 1 − A2(τ) (64) 9 where we have introduced the integral I(τ) ≡ τZ −τ0 dτ ′eiλ(τ ′)A(τ ′)
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Hidden facts in Landau-Zener transitions revealed by the Riccati Equation
The Riccati equation bridges the two probability amplitudes in Landau-Zener transitions and pinpoints why the Markov approximation succeeds for one amplitude but fails for the other.