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Concept
Adiabatic theorem
Summary
The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:
A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.
In simpler terms, a quantum mechanical system subjected to gradually changing external conditions adapts its functional form, but when subjected to rapidly varying conditions there is insufficient time for the functional form to adapt, so the spatial probability density remains unchanged.
At some initial time a quantum-mechanical system has an energy given by the Hamiltonian ; the system is in an eigenstate of labelled . Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian at some later time . The system will evolve according to the time-dependent Schrödinger equation, to reach a final state . The adiabatic theorem states that the modification to the system depends critically on the time during which the modification takes place.
For a truly adiabatic process we require ; in this case the final state will be an eigenstate of the final Hamiltonian , with a modified configuration:
The degree to which a given change approximates an adiabatic process depends on both the energy separation between and adjacent states, and the ratio of the interval to the characteristic time-scale of the evolution of for a time-independent Hamiltonian, , where is the energy of .
Conversely, in the limit we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged:
The so-called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the spectrum of is discrete and nondegenerate, such that there is no ambiguity in the ordering of the states (one can easily establish which eigenstate of corresponds to ). In 1999 J. E. Avron and A. Elgart reformulated the adiabatic theorem to adapt it to situations without a gap.
Official source
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