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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).
The model can answer questions such as the probability that this occurs within finite time, or the mean time until which it occurs. First-hitting-time models can be applied to expected lifetimes, of patients or mechanical devices. When the process reaches an adverse threshold state for the first time, the patient dies, or the device breaks down.
The most important inference derived from the steady state equation and the equation for fractional change over time is that the elimination rate constant (k e) or the sum of rate constants that apply in a model determine the time course for change in mass when a system is perturbed (either by changing the rate of inflow or production, or by ...
In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time. Crossing states that the same formula that determines the S-matrix elements and scattering amplitudes for particle to scatter with and produce particle and ...
The effect is named after Wilhelm Hanle, who was the first to explain the effect, in terms of classical physics, in Zeitschrift für Physik in 1924. [ 8 ] [ 9 ] Initially, the causes of the effect were controversial, and many theorists mistakenly thought it was a version of the Faraday effect .
From the equations, the power series must start with at least an order of to satisfy the real part of the equation; for a good classical limit starting with the highest power of the Planck constant possible is preferable, which leads to = = and = = (), with the following constraints on the lowest order terms, () = (()) and () =
When φ(t) is constrained to its principal value, either the interval (−π, π] or [0, 2π), it is called wrapped phase. Otherwise it is called unwrapped phase, which is a continuous function of argument t, assuming s a (t) is a continuous function of t. Unless otherwise indicated, the continuous form should be inferred. Instantaneous phase ...