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A bilayer nano-stanene-like structure with Ruderman–Kittel–Kasuya–Yoshida (RKKY) coupling described by the Ising model is proposed. The magnetic and thermodynamic properties are studied using the effective-field theory with correlations. The exchange coupling, longitudinal magnetic field, number of non-magnetic layers, and anisotropies had major influences on the magnetization, specific heat, and internal energy. Different saturation magnetizations are observed on the magnetization curve. The variation in the system blocking temperature is studied. The results provide theoretical guidance for the magnetic investigation of nanomaterials with RKKY coupling.

On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is developed and used to study the evolution of a quantum scalar field in Robertson-Walker flat spacetime. We first solve the functional Schrodinger equation for a free scalar field and obtain the exact solutions, of which the 'ground-state' solution possesses isotropy and homogeneity automatically We then investigate the way of extending the method to treat the case in which there is a high-order perturbative self-interaction.

The invariant formulation is used to investigate the quantum motion of an ion in a combined trap. The classical correspondence of the quantum motion and stability problem are then discussed.

A path integral method is used to study the time evolution of a three-dimensional time-dependent system. The propagator for a charged particle in a time-dependent magnetic field and quadrupolar electric potential is obtained.

In this paper, the exact evolving state for a two-level quantum system is found by making use of the Lewis-Riesenfeld invariant theory. In principle, the correction to an arbitrary order in the adiabatic approximation parameter can be obtained from this exact evolving state. To the -th order in , it is shown that the exact evolving state reduces to the superadiabatic basis introduced by Berry recently. The Aharonov-Anandan phase and its adiabatic limit (Berry phase for the system) are also calculated.

The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.