q形变邓范模型中的量子信息分析
Quantum Information Analysis in a q-Deformed Deng-Fan Model
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中文总结 AI 辅助
本文提出q形变邓范势（qDFP）模型，精确求解其薛定谔方程，结合信息论度量分析量子信息分布，揭示形变参数对光谱、束缚态及熵不确定关系的影响，q→1时退化为标准邓范势。
中文摘要 AI 辅助
我们提出了一种q形变邓范势（qDFP）模型，该模型可在保持平衡构型的同时对短程排斥和长程吸引进行可控调制。我们在定态薛定谔方程框架内精确求解该模型，得到了以超几何函数表示的能量本征值和波函数的闭式表达式。研究表明，形变参数q会诱导非均匀的光谱偏移和束缚态的重新分布；特别地，当q<1时，系统呈现光谱压缩和空间局域化增强的特性。此外，我们从信息论视角，在位置空间和动量空间中分别利用香农熵、费舍尔信息及费舍尔-香农复杂度度量对该系统展开研究。结果显示，形变参数支配着量子信息的重新分布，且根据Białynicki-Birula和Mycielski（BBM）熵不确定原理，建立了空间局域化与动量非局域化之间的直接关联。更强的形变会使量子态进一步偏离最小不确定构型，导致BBM界之上的熵过剩增加，互补可观测量的信息含量降低，即便位置空间的局域化程度有所提升。对熵密度和费舍尔信息密度的分析进一步表明，形变如何重塑量子态的局域信息含量与结构复杂度。我们还证明，在q→1的极限下，qDFP模型会退化为标准邓范势。
英文摘要
We introduce a $q$-deformed Deng-Fan potential ($q$DFP) model that enables controlled modulation of short-range repulsion and long-range attraction while preserving the equilibrium configuration. The model is solved exactly within the framework of the time-independent Schrödinger equation, yielding closed-form expressions for the energy eigenvalues and wave functions in terms of hypergeometric functions. We show that the deformation parameter $q$ induces non-uniform spectral shifts and a redistribution of bound states. In particular, for $q<1$, the system exhibits spectral compression and enhanced spatial localization. In addition, we investigate the system from an information-theoretic perspective using Shannon entropy, Fisher information, and Fisher--Shannon complexity measures in both position and momentum spaces. The results reveal that the deformation parameter governs the redistribution of quantum information, establishing a direct connection between spatial confinement and momentum delocalization in accordance with the Białynicki-Birula and Mycielski entropic uncertainty principle. Stronger deformation pushes the quantum state further from the minimum-uncertainty configuration, increasing the entropic excess above the BBM bound and reducing the information content about complementary observables, even as position-space localization sharpens. The analysis of entropic and Fisher information densities further shows how the deformation reshapes both the local information content and the structural complexity of the quantum states. We show that in the limit $q\to 1$, the $q$DFP model is reduced to the standard Deng-Fan potential.
