Derivation of the Landau-Zener formula via functional equations

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.