[论文解读] Comparative analysis of electric field influence on the quantum wells with different boundary conditions. I. Energy spectrum, quantum information entropy and polarization

O. Olendski|PubMed|Feb 10, 2015

Cold Atom Physics and Bose-Einstein Condensates参考文献 80被引用 9

一句话总结

本文对在垂直电场作用下具有狄利克雷与诺伊曼边界条件所有组合的一维量子阱进行了分析与数值比较研究。研究发现，电场通过场驱动的能级混合，使基态与激发态的极化方向相反；并表明极化、能谱及量子信息熵强烈依赖于边界条件，精确解证实了赫尔曼-费曼定理与熵不确定关系。

ABSTRACT

Analytical solutions of the Schrödinger equation for the one-dimensional quantum well with all possible permutations of the Dirichlet and Neumann boundary conditions (BCs) in perpendicular to the interfaces uniform electric field [Formula: see text] are used for the comparative investigation of their interaction and its influence on the properties of the system. Limiting cases of the weak and strong voltages allow an easy mathematical treatment and its clear physical explanation; in particular, for the small [Formula: see text], the perturbation theory derives for all geometries a linear dependence of the polarization on the field with the BC-dependent proportionality coefficient being positive (negative) for the ground (excited) states. Simple two-level approximation elementary explains the negative polarizations as a result of the field-induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role. Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically. Hellmann-Feynman theorem reveals a fundamental relation between the polarization and the speed of the energy change with the field. It is proved that zero-voltage position entropies [Formula: see text] are BC independent and for all states but the ground Neumann level (which has [Formula: see text]) are equal to [Formula: see text] while the momentum entropies [Formula: see text] depend on the edge requirements and the level. Varying electric field changes position and momentum entropies in the opposite directions such that the entropic uncertainty relation is satisfied. Other physical quantities such as the BC-dependent zero-energy and zero-polarization fields are also studied both numerically and analytically. Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.

研究动机与目标

• 研究垂直电场对具有狄利克雷与诺伊曼边界条件所有排列组合的量子阱的影响。
• 分析边界条件如何影响电场存在下能谱、极化及量子信息熵。
• 通过赫尔曼-费曼定理建立极化与能量随场强变化率之间的基本联系。
• 探讨位置与动量熵的行为，并验证不同边界条件下熵不确定关系的成立。

提出的方法

• 在一维薛定谔方程中，对具有混合狄利克雷与诺伊曼边界条件的系统在均匀电场下求解解析解。
• 对弱电场应用微扰理论，推导出与边界条件相关的系数下，极化随场强的线性依赖关系。
• 采用两能级近似，解释负极化源于相邻未扰动态之间的相消干涉。
• 对极限情况（弱场：微扰区；强场：一堵墙失效的渐近区）进行数值与解析处理。
• 评估位置与动量信息熵 $S_x$ 与 $S_k$，以评估不确定性并验证熵不确定关系。
• 应用赫尔曼-费曼定理，将能量对场的导数与极化关联起来。

实验结果

研究问题

• RQ1 电场如何影响具有不同狄利克雷与诺伊曼边界条件组合的量子阱的能谱？
• RQ2 为何在弱电场下激发态的极化变为负值，且其行为如何依赖于边界条件？
• RQ3 能级混合（特别是与邻近态的混合）在诱导负极化中起何作用？
• RQ4 随着电场增强，位置与动量熵如何演化，熵不确定关系是否保持成立？
• RQ5 赫尔曼-费曼定理揭示的极化与能量随场强变化率之间的基本关系是什么？

主要发现

• 在弱电场下，基态极化随场强线性增加，而激发态由于场驱动的邻近能级混合表现出负极化。
• 弱场下极化的大小依赖于边界条件，基态为正值，激发态为负值，该结果经解析与数值方法共同验证。
• 两能级近似表明，仅相邻态的混杂主导了极化变化，解释了负极化的起源为相消干涉。
• 位置熵 $S_x$ 随场强增加而减小（表明局域化增强），而动量熵 $S_k$ 增加，熵不确定关系得以保持。
• 在零场时，除诺伊曼基态外，所有态的位置熵 $S_x$ 均与边界条件无关，且等于 $\ln 2 - 1 \approx -0.30686$，而诺伊曼基态的 $S_x = 0$；$S_k$ 同时依赖于边界条件与能级。
• 在强场极限下，量子阱有效简化为单壁系统，其解仅由粒子被推向的那堵墙的边界条件决定。