Abstract

We obtained the energy and wave functions of a particle in a quantum corral subjected to a constant magnetic field, as a function of the radius of the quantum corral \(R_\mathrm{c}\) and the intensity of the magnetic field \(b^2\). We also computed the standard deviation and the Shannon information entropies as a function of \(R_\mathrm{c}\) and \(b^2\), which in turn are compared to determine their effectiveness in measuring particle (de)localization. For a fixed magnitude of the magnetic field \(b^2\), the Shannon entropy of all states diminishes as the confinement radius \(R_\mathrm{c}\) decreases revealing an extensive localization. For a fixed value of \(R_\mathrm{c}\), the Shannon entropy of the states (0, 0) and (0, 1) decreases monotonically as the magnetic field \(b^2\) grows, whereas for the states (1, 0), (2, 0), (1, 1) and (2, 1), the Shannon entropy grows slowly, reaching a maximum (delocalization), and then diminishes as \(b^2\) increases. The expectation value of \(\left\langle r\right\rangle \) for a fixed value \(R_\mathrm{c}\), for the states (0, 0) and (0, 1), decreases monotonically as \(b^2\) increases, whereas for the states (1, 0), (2, 0), (1, 1) and (2, 1) increases and after reaching a maximum, it decreases as \(b^2\) grows. This behavior is counter-intuitive because the particle is forecasted to be closer to the origin as the magnetic field grows.

Graphic abstract

The Shannon entropy of few low lying states of a quantum corral as a function of magnetic field.

中文翻译：

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在恒定磁场存在下，粒子在量子皮质中的局域化-局域化

摘要

我们获得了处于恒定磁场的量子畜栏中粒子的能量和波函数，该函数是量子畜栏半径\（R_ \ mathrm {c} \）和磁场强度\（ b ^ 2 \）。我们还计算了标准偏差和Shannon信息熵作为\（R_ \ mathrm {c} \）和\（b ^ 2 \）的函数，然后将它们进行比较以确定它们在测量粒子（去定位）方面的有效性。 。对于固定大小的磁场\（b ^ 2 \），随着约束半径\（R_ \ mathrm {c} \）的减小，所有状态的香农熵都减小，这表明存在广泛的局域性。对于固定值\（R_ \ mathrm {c} \），随着磁场\（b ^ 2 \）的增长，状态（0，0）和（0，1 ）的香农熵单调降低，而对于状态（1， 0），（2，0），（1、1）和（2，1）时，香农熵缓慢增长，达到最大值（离域），然后随着\（b ^ 2 \）的增加而减小。对于状态为（0，0）和（0，1 ）的固定值\（R_ \ mathrm {c} \）的\（\ left \ langle r \ right \ rangle \）的期望值单调递减为\（b ^ 2 \）增加，而对于状态（1，0），（2，0），（1、1）和（2，1）增加，并且在达到最大值后，它随着\（b ^ 2 \）成长。这种行为是违反直觉的，因为随着磁场的增长，预计粒子将更靠近原点。

图形摘要

量子畜栏的少数低位态的香农熵与磁场的函数关系。