Adiabatic theorem

The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:

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.

Diabatic process: Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the spatial probability density remains unchanged. Typically there is no eigenstate of the final Hamiltonian with the same functional form as the initial state. The system ends in a linear combination of states that sum to reproduce the initial probability density.

Adiabatic process: Gradually changing conditions allow the system to adapt its configuration, hence the probability density is modified by the process. If the system starts in an eigenstate of the initial Hamiltonian, it will end in the corresponding eigenstate of the final Hamiltonian.

At some initial time ${\displaystyle \scriptstyle {t_{0}}}$ a quantum-mechanical system has an energy given by the Hamiltonian ${\displaystyle \scriptstyle {{\hat {H}}(t_{0})}}$; the system is in an eigenstate of ${\displaystyle \scriptstyle {{\hat {H}}(t_{0})}}$ labelled ${\displaystyle \scriptstyle {\psi (x,t_{0})}}$. Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian ${\displaystyle \scriptstyle {{\hat {H}}(t_{1})}}$ at some later time ${\displaystyle \scriptstyle {t_{1}}}$. The system will evolve according to the time-dependent Schrödinger equation, to reach a final state ${\displaystyle \scriptstyle {\psi (x,t_{1})}}$. The adiabatic theorem states that the modification to the system depends critically on the time ${\displaystyle \scriptstyle {\tau =t_{1}-t_{0}}}$ during which the modification takes place.

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