For the exactly solvable model of a two-mode oscillator with slowly varying frequency of one of the modes it is demonstrated that such a quantum number as the angular momentum of the oscillator is not conserved in the adiabatic limit.
The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.
By stating the adiabatic theorem of quantum mechanics in a clear and rigorous way, we establish a necessary condition and a sufficient condition for its validity, where the latter is obtained employing our recently developed adiabatic perturbation theory. Also, we simplify further the sufficient condition into a useful and simple practical test at the expense of its mathematical rigor. We present results for the most general case of quantum systems, i.e., those with degenerate energy spectra. These conditions are of upmost importance for assessing the validity of practical implementations of non-Abelian braiding and adiabatic quantum computation. To illustrate the degenerate adiabatic approximation, and the necessary and sufficient conditions for its validity, we analyze in depth an exactly solvable time-dependent degenerate problem.
In a quantum-mechanical problem with a slowly varying Hamiltonian, such as appears in a semiclassical treatment of inelastic atomic collisions, it is common to restrict attention to a small number of states among which transitions are anticipated. This procedure is justified by an extension of the conventional adiabatic theorem to nearly degenerate states. The correct mixing within the restricted basis is determined by a time-dependent variational principle. The contribution of states outside the basis vanishes as the time scale for variation of the Hamiltonian becomes large. A further generalization deals with pseudoeigenfunctions, such as appear in the theory of the Stark effect.
The evolution mod Psi (t)) of a system with slowly-varying Hamiltonian H( Omega t) depends not only on the slowness parameter Omega but also on Planck's constant h(cross). For systems with two (or more) classically separated phase-space regions which are quantally connected by tunnelling, the curves of instantaneous energy levels display hyperbolic near-degeneracies rather than crossings. In such cases the limits Omega to 0 (h(cross) fixed and small) and h(cross) to 0 ( Omega fixed and small) lead to opposite behaviour of mod Psi (t)). As an illustration, the uniform semiclassical adiabatic behaviour ( Omega and h(cross) small, Omega /h(cross) arbitrary) is calculated exactly for a double-well potential for which one well gets shallower as the other gets deeper.