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We analyze the validity of the adiabatic approximation, and in particular the reliability of what has been called the standard criterion for validity of this approximation. Recently, this criterion has been found to be insufficient. We will argue that the criterion is sufficient only when it agrees with the intuitive notion of slowness of evolution of the Hamiltonian. However, it can be insufficient in cases where the Hamiltonian varies rapidly but only by a small amount. We also emphasize the distinction between the adiabatic {em theorem} and the adiabatic {em approximation}, two quite different although closely related ideas.
Recently, the authors of Ref.1[arXiv:1004.3100] claimed that they have proven the traditional adiabatic condition is a necessary condition. Here, it is claimed that there are some mistakes and an artificial over-strong constraint in [1], making its result inconvincible.
Recently there have been some controversies about the criterion of the adiabatic approximation. It is shown that an approximate diagonalization of the effective Hamiltonian in the second quantized formulation gives rise to a reliable and unambiguous
We investigate the origin of quantum geometric phases, gauge fields and forces beyond the adiabatic regime. In particular, we extend the notions of geometric magnetic and electric forces discovered in studies of the Born-Oppenheimer approximation to
We study the adiabatic-impulse approximation (AIA) as a tool to approximate the time evolution of quantum states, when driven through a region of small gap. The AIA originates from the Kibble-Zurek theory applied to continuous quantum phase transitio
This paper is devoted to a generalisation of the quantum adiabatic theorem to a nonlinear setting. We consider a Hamiltonian operator which depends on the time variable and on a finite number of parameters and acts on a separable Hilbert space of whi