Anti-adiabatic evolution in quantum-classical hybrid system

The adiabatic theorem is an important concept in quantum mechanics, it tells that a quantum system subjected to gradually changing external conditions remains to the same instantaneous eigenstate of its Hamiltonian as it initially in. In this paper, we study the another extreme circumstance where the external conditions vary rapidly such that the quantum system can not follow the change and remains in its initial state (or wavefunction). We call this type of evolution anit-adiabatic evolution. We examine the matter-wave pressure in this situation and derive the condition for such an evolution. The study is conducted by considering a quantum particle in an infinitely deep potential, the potential width $Q$ is assumed to be change rapidly. We show that the total energy of the quantum subsystem decreases as $Q$ increases, and this rapidly change exerts a force on the wall which plays the role of boundary of the potential. For $Q<Q_{0}$ ($Q_0$ is the initial width of the potential), the force is repulsive, and for $Q>Q_{0}$, the force is positive. The condition for the anti-adiabatic evolution is given via a spin-$\frac 1 2$ in a rotating magnetic field.