Adiabatic theorem

The adiabatic theorem of quantum mechanics, also called Adiabatensatz quantum mechanics indicates that the state of the system

Good approximation remains in the -th eigenstate of the course of the adiabatic time evolution, when the Hamiltonian of a system " slow enough " changes (for example due to external influences).

"Slow enough " means ( for ) that

Applies.

It is the characteristic time of the transition of the system from state to state, and the states and the energy and associated eigenvalues ​​of the system.

This means that the change of is slow compared to the natural time scale of the system, which is defined by transitions between the energy eigenstates.

In the adiabatic limit case are the changes infinitely slowly:

Examples of physics

The best-known example in physics is the Born- Oppenheimer approximation. Max Born and Robert Oppenheimer showed that for the calculation of the changes of state of electrons in a molecule, the motion of the atomic nuclei (the change of ) can be neglected. Simply put, the electrons move so fast and the time they need for a transition between two electron levels is so short that the movement of the ( slow ) nuclei is irrelevant for computation.

History

The adiabatic theorem of quantum mechanics goes back to works by Max Born and Vladimir Aleksandrovich Fock from the year 1928. A complete mathematical formulation, however, it was not until Tosio Kato (1950 ) in connection with the perturbation theory of linear operators.

Michael Berry showed in 1984 that in case of cyclic adiabatic change of parameters the system will still return to its initial state, but under certain circumstances a dependent on the geometry of the parameter space phase factor is obtained ( Berry phase ).