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In quantum physics and quantum chemistry, an avoided crossing (AC, sometimes called intended crossing, [1] non-crossing or anticrossing) is the phenomenon where two eigenvalues of a Hermitian matrix representing a quantum observable and depending on continuous real parameters cannot become equal in value ("cross") except on a manifold of dimension . [2]
Sketch of an avoided crossing.The graph represents the energies of the system along a parameter z (which may be vary in time). The dashed lines represent the energies of the diabatic states, which cross each other at z c, and the full lines represent the energy of the adiabatic states (eigenvalues of the Hamiltonian).
Traditionally, comparative results in economics are obtained using the Implicit Function Theorem, an approach that requires the concavity and differentiability of the objective function as well as the interiority and uniqueness of the optimal solution. The methods of monotone comparative statics typically dispense with these assumptions.
The single-crossing condition is also used in applications where there are a few agents or types of agents that have preferences over an ordered set. Such situations appear often in information economics , contract theory , social choice and political economics , among other fields.
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Here’s how far in a crosswalk a pedestrian has to be before you can drive through it, and what the fine is for not yielding the right of way.
Here are some ways to get around several major crossings. You can’t spend much time in Columbia without a train blocking your path. Here are some ways to get around several major crossings.
Ideal Conical Intersection. In quantum chemistry, a conical intersection of two or more potential energy surfaces is the set of molecular geometry points where the potential energy surfaces are degenerate (intersect) and the non-adiabatic couplings between these states are non-vanishing.