SWhat changed
Two interacting quantum energy levels can avoid crossing even when their uncoupled energies would meet. Zener solved an ideal two-state crossing model to show how the rate of change affects the chance of a non-adiabatic transition. With constant coupling V and a linear uncoupled energy difference of slope s, the asymptotic probability is exp(-2*pi*V^2/(hbar*abs(s))). Slow and fast passages can therefore have very different outcomes.
Scientific scope
Two interacting quantum energy levels can avoid crossing even when their uncoupled energies would meet. Zener solved an ideal two-state crossing model to show how the rate of change affects the chance of a non-adiabatic transition. With constant coupling V and a linear uncoupled energy difference of slope s, the asymptotic probability is exp(-2*pi*V^2/(hbar*abs(s))). Slow and fast passages can therefore have very different outcomes. The exact formula belongs to the ideal all-time linear crossing with fixed coupling and basis states. Applying it to a molecular collision requires the trajectory to be treated as an external parameter and the transition region to justify those approximations. It is not an unrestricted finite-time, noisy or multilevel result. Landau, Stueckelberg and Majorana retain their separate historical contributions; the conditional comparison with Landau reported through an intermediary in this paper is not independent evidence of an error in Landau original. Fowler communicated the paper and is not a coauthor.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The exact formula belongs to the ideal all-time linear crossing with fixed coupling and basis states. Applying it to a molecular collision requires the trajectory to be treated as an external parameter and the transition region to justify those approximations. It is not an unrestricted finite-time, noisy or multilevel result. Landau, Stueckelberg and Majorana retain their separate historical contributions; the conditional comparison with Landau reported through an intermediary in this paper is not independent evidence of an error in Landau original. Fowler communicated the paper and is not a coauthor.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.5. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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