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arXiv 2411.04033math-phmath.MPphysics.class-phquant-ph
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基于薛定谔波函数的自由欧拉-伯努利梁中的能量传输

Energy transport in a free Euler-Bernoulli beam in terms of Schrödinger's wave function

Serge N. Gavrilov, Anton M. Krivtsov, Ekaterina V. Shishkina

备注：
5 pages
发表信息：
Mechanics Research Communications 144 (2025) 104382
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AI总结：

本文研究一维自由欧拉-伯努利梁方程与含时薛定谔方程的等价性，建立了梁的应变与粒子速度对同薛定谔波函数的一一对应关系，并证明梁的机械能密度与波函数的概率密度具有完全相同的传播方式。

摘要：

薛定谔方程在经典力学框架中并不常用，尽管从历史上看，该方程是作为简化方程推导出来的，它与经典的 Germain-Lagrange 动力板方程等价。关于这种等价的确切含义问题在现代文献中仍有讨论。在本文中，我们考虑一维情况，此时 Germain-Lagrange 方程退化为欧拉-伯努利方程，后者被用于经典的梁理论中。我们建立了一一对应关系，该关系存在于具有任意复值初始数据的一维含时薛定谔方程（针对自由粒子）的所有解（即波函数 $ψ$）集合，与有序量对（梁应变和粒子速度）集合之间，这些量对表征了具有任意实值初始数据的梁方程的解 $u$。因此，自由无限欧拉-伯努利梁的动力学可以由自由粒子的薛定谔方程来描述，反之亦然。最后，我们证明，对于两个对应的解 $u$ 和 $ψ$，为 $u$ 计算的机械能密度在梁中的传播方式，与为 $ψ$ 计算的概率密度的传播方式完全相同。

英文摘要：

The Schrödinger equation is not frequently used in the framework of the classical mechanics, though historically this equation was derived as a simplified equation, which is equivalent to the classical Germain-Lagrange dynamic plate equation. The question concerning the exact meaning of this equivalence is still discussed in modern literature. In this note, we consider the one-dimensional case, where the Germain-Lagrange equation reduces to the Euler-Bernoulli equation, which is used in the classical theory of a beam. We establish a one-to-one correspondence between the set of all solutions (i.e., wave functions $ψ$) of the 1D time-dependent Schrödinger equation for a free particle with arbitrary complex valued initial data and the set of ordered pairs of quantities (the beam strain and the particle velocity), which characterize solutions $u$ of the beam equation with arbitrary real valued initial data. Thus, the dynamics of a free infinite Euler-Bernoulli beam can be described by the Schrödinger equation for a free particle and vice versa. Finally, we show that for two corresponding solutions $u$ and $ψ$ the mechanical energy density calculated for $u$ propagates in the beam exactly in the same way as the probability density calculated for $ψ$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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