Landau–Zener formula¶
An analytic two-level quantum-transition probability for a constant-coupling avoided crossing whose diabatic energy separation varies linearly in time from the remote past to the remote future.
Core Idea¶
The Landau–Zener formula gives an asymptotic nonadiabatic transition probability for an isolated two-state quantum Hamiltonian whose diabatic energy separation changes linearly in time and whose coupling is constant. A system prepared far before the crossing evolves through the minimum gap and is read far afterward. A system prepared far before the crossing evolves through the minimum gap and is read far afterward.
Scope of Application¶
The formula applies to controlled or approximated isolated two-level crossings in atomic, molecular, condensed-matter, spin, and quantum-control settings. Use it only after fixing basis and probability convention, sweep slope, coupling, initial/final limits, and the relevance of extra levels, nonlinear driving, finite time, noise, and decoherence.
- Atomic crossings. Estimates diabatic transition probability.
- Spin sweeps. Models a level passage under changing field.
- Molecular collisions. Approximates local two-state crossings with care.
- Quantum annealing. Diagnoses gap/sweep tradeoffs.
- Multistate theory. Provides a local building block when extension is justified.
Clarity¶
A result should state basis, initial state, target event, gap/coupling convention, detuning slope, time interval, and whether the reported probability is adiabatic or diabatic. Opposite naming conventions can produce complementary probabilities without disagreement. The closest near miss sets the boundary: A generic avoided crossing is the nearest miss: it supplies the spectral geometry but not the linear sweep, constant coupling, and asymptotic conditions required for the formula.
Manages Complexity¶
The formula compresses a time-dependent Schrödinger evolution into one dimensionless competition between coupling and sweep. That compression supports design intuition while concealing phase, interference, environment, and multilevel structure. The central slow adiabaticity–fast control tradeoff is this: Slow passage suppresses nonadiabatic excitation while increasing exposure time and cost. A second two-level tractability–multilevel fidelity tension matters because A local pair admits a closed form while neighboring states may participate. The asymptotic exactness–finite experiment tension adds that The standard solution assumes remote endpoints unavailable in practice.
Abstract Reasoning¶
Use three linked moves: reduce the relevant spectrum to a justified two-state subspace; linearize diabatic detuning near the crossing and test the time window; estimate the coupling and confirm it is effectively constant. As a collapse test, the case exits when other levels, decoherence, noise, nonlinear detuning, or finite-time effects materially control the transition without an explicit corrected model. A fourth check is to map preparation and readout to the chosen adiabatic or diabatic convention. A final check is to use the formula only after checking decoherence, extra levels, and finite-time corrections.
Knowledge Transfer¶
The avoided-crossing calculation transfers among physical platforms when Hamiltonians reduce to the same two-level form. A social or optimization ‘crossing’ lacks quantum amplitude dynamics and is analogy only. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. The formula quantifies change between energy-state descriptions. Sweep time and nonadiabatic probability oppose one another under the model.
Neighborhood in Abstraction Space¶
Landau–Zener formula sits in a sparse region of the domain-specific corpus (71st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Dynamical Systems & Differential Structures (37 abstractions)
Nearest neighbors
- Lieb–Liniger model — 0.86
- Level Repulsion — 0.83
- Translation operator (quantum mechanics) — 0.83
- Bogdanov–Takens bifurcation — 0.83
- Heteroclinic Cycle — 0.83
Computed from structural-signature embeddings · 2026-10-08