In the ballistic regime we study both semiclassically and quantum mechanically the electron’s dynamics in two-dimensional electron gas in the presence of an inhomogeneous magnetic field applied perpendicular to the plane. The magnetic field is constant inside four separate circular regions which are located at the four corners of a square of side length larger than the diameter of the circles, while outside the circles the magnetic field is zero. We carry out the stability analysis of the periodic orbits and for given initial conditions numerically calculate the two-dimensional invariant torus embedded in the four-dimensional phase space. Applying the Bohr–Sommerfeld and the Einstein–Brillouin–Keller semiclassical quantization methods we obtain the energy levels for different magnetic field strengths. We also perform exact quantum calculations solving numerically the discretized version of the Schrdinger equation. In our calculations, we consider only those bound states that are localized to the neighborhood of the four magnetic disks. We show that the semiclassical results are in good agreement with those found from our quantum calculations. Moreover, the current distribution and the phase of the different wave functions enable us to deduce the two quantum numbers n ₁ and n ₂ characterizing the energy levels in the semiclassical methods. Finally, we present two examples in which the quantum state shows a similar structure to the previous states, but these are special in the following sense. One of them is a scar state localized to the neighborhood of the periodic orbit while this orbit is already unstable. In the case of the other state, the current density is circulating in two rings in opposite directions. Thus, it is not consistent with the classical motion in the neighborhood of the periodic orbit.

在弹道系统中，我们从半经典和量子力学上研究了在垂直于平面施加不均匀磁场的情况下，二维电子气中电子的动力学。磁场在四个独立的圆形区域内是恒定的，这些区域位于边长大于圆直径的正方形的四个角上，而在圆外磁场为零。我们对周期轨道进行稳定性分析，并在给定的初始条件下数值计算嵌入在四维相空间中的二维不变环面。应用 Bohr-Sommerfeld 和 Einstein-Brillouin-Keller 半经典量化方法，我们获得了不同磁场强度的能级。我们还执行精确的量子计算，以数值方式求解 Schrdinger 方程的离散版本。在我们的计算中，我们只考虑定位在四个磁盘附近的那些束缚态。我们表明半经典结果与我们从量子计算中发现的结果非常一致。此外，不同波函数的电流分布和相位使我们能够推导出两个量子数n ₁和n ₂表征半经典方法中的能级。最后，我们展示了两个例子，其中量子态显示出与之前的状态相似的结构，但这些在以下意义上是特殊的。其中之一是局部于周期轨道附近的疤痕状态，而该轨道已经不稳定。在另一种状态的情况下，电流密度在两个相反方向的环中循环。因此，它与周期轨道附近的经典运动不一致。