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        "description": "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.",
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      "statement": "# Zener: transition law for an ideal linear two-level crossing\n\nTwo 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.\n\n## Scientific scope\n\nThe 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.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nClarence Zener\n\nSource-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.\n\n\n\n## Original sources\n\n- [Non-adiabatic crossing of energy levels (original1932scan reproduced as Wittig2025AppendixI.4.1)](https://curtwittig.com/wp-content/uploads/2025/05/Notes-Project-CH-I.4-Final-5.13.25.pdf)\n",
      "summary": "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.",
      "title": "Zener: transition law for an ideal linear two-level crossing"
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    "title": "Zener: transition law for an ideal linear two-level crossing",
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    "what_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.",
    "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.",
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