Derivation of the Landau-Zener formula via functional equations
arXiv:2504.02576 · doi:10.1088/1751-8121/ae0219
Abstract
The Landau-Zener formula describes the diabatic transition probability of a two-level system under linear driving. Its rigorous derivation typically relies on sophisticated mathematical tools, such as special functions, Laplace transforms, or contour integrals. In this work, we present a derivation of the Landau-Zener transition probability using a fundamentally different approach via functional equations. By leveraging integrability, we prove that this transition probability satisfies a functional equation, whose solutions establish the exponential form of the formula. The coefficient in the exponent is then determined through a lowest-order perturbation calculation. This derivation is rigorous and does not involve any sophisticated mathematics. Our work provides insights into the origin of the exponential form of the Landau-Zener transition probability, and shows that the Landau-Zener formula can be viewed as a consequence of integrability, though the two-level Landau-Zener Hamiltonian itself does not satisfy the integrability conditions.
arXiv comment9 pages, 1 figure; discussions expanded, summary of other derivation methods added; version accepted as Letter in Journal of Physics A: Mathematical and Theoretical
References in corpus (15)
- Landau-Zener-Stuckelberg interferometry
- Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference
- Landau-Zener transition in a multilevel system. An exact result
- Integrable time-dependent quantum Hamiltonians
- Counterintuitive transitions in the multistate Landau-Zener problem with linear level crossings
- Quantum integrability in systems with finite number of levels
- Integrable time-dependent Hamiltonians, solvable Landau-Zener models and Gaudin magnets
- Majorana's approach to nonadiabatic transitions validates the adiabatic-impulse approximation
- Solvable 4-state Landau-Zener model of two interacting qubits with path interference
- Exact transition probabilities in a six-state Landau-Zener system with path interference
- Exact results for models of multichannel quantum nonadiabatic transitions
- Quantum integrability in the multistate Landau-Zener problem
- Comment on "Exact results for survival probability in the multistate Landau-Zener model"
- Dynamic spin localization and gamma-magnets
- Multitime Landau-Zener model: classification of solvable Hamiltonians