antikrossing

Anticrossing, also called avoided crossing, is a phenomenon in quantum mechanics where two energy levels that would cross as a parameter is varied instead repel each other and acquire a minimum separation. This occurs when the states connected to the levels are coupled, causing mixing of the states and a breakdown of a true degeneracy at the would-be crossing point.

In a minimal two-level model, the Hamiltonian can be written as H = [ε1, V; V, ε2], where ε1 and ε2 are the energies of the uncoupled states and V is the coupling between them. The eigenvalues are E± = (ε1 + ε2)/2 ± sqrt(((ε1 − ε2)/2)^2 + V^2). When ε1 ≈ ε2, the square-root term is dominated by V, producing a gap of 2|V|. The corresponding eigenstates become superpositions of the original basis states, reflecting the mixing induced by the coupling.

The presence of an anticrossing implies that symmetry or selection rules allow the coupling V; if the coupling is forbidden, the levels may cross without interaction. Anticrossings are observed across various domains, including solid-state physics where hybridization of electronic bands creates gaps, quantum dots and superconducting qubits, and molecular spectroscopy where potential energy surfaces exhibit avoided crossings as nuclear geometry changes. In chemistry, avoided crossings are related to the broader concept of conical intersections in higher-dimensional systems. The related Landau–Zener framework describes the transition probability between diabatic states as the system traverses the avoided crossing under time-dependent variation.