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首页> 外文期刊>Physica, A. Statistical mechanics and its applications >Tsallis' maximum entropy ansatz leading to exact analytical time dependent wave packet solutions of a nonlinear Schr?dinger equation
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Tsallis' maximum entropy ansatz leading to exact analytical time dependent wave packet solutions of a nonlinear Schr?dinger equation

机译：Tsallis的最大熵ansatz导致非线性Schr？dinger方程的精确解析时变波包解

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摘要

Tsallis maximum entropy distributions provide useful tools for the study of a wide range of scenarios in mathematics, physics, and other fields. Here we apply a Tsallis maximum entropy ansatz, the q-Gaussian, to obtain time dependent wave-packet solutions to a nonlinear Schr?dinger equation recently advanced by Nobre, Rego-Monteiro and Tsallis (NRT) [F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Phys. Rev. Lett. 106 (2011) 140601]. The NRT nonlinear equation admits plane wave-like solutions (q-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the q-generalized thermostatistical formalism, is characterized by a parameter q and in the limit q→1 reduces to the standard, linear Schr?dinger equation. The q-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known q-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the q→1 limit the q-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schr?dinger equation. In the present work we also show that there are other families of nonlinear Schr?dinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.
机译：Tsallis最大熵分布为研究数学，物理和其他领域的各种情况提供了有用的工具。在这里，我们应用Tsallis最大熵ansatz q-Gaussian来获得由Nobre，Rego-Monteiro和Tsallis（NRT）最近提出的非线性Schr？dinger方程的时变波包解。 Nobre，M.A. Rego-Monteiro，C.Tsallis，物理学牧师106（2011）140601]。 NRT非线性方程允许平面波状解（q平面波）与著名的de Broglie关系兼容，该关系将波数和频率分别与能量和动量联系起来。 NRT方程受q广义热统计形式主义的启发，其特征在于参数q，在极限q→1内可简化为标准的线性薛定er方程。在此研究的NRT方程的q-高斯解可以作为一个特定实例来接受先前已知的q平面波解。因此，当前的工作扩展了NRT动力学产生的可能的过程范围，这些过程允许进行分析，精确的处理。在q→1极限中，q-高斯解对应于自由粒子线性薛定ding方程的高斯波包解。在本工作中，我们还表明，除了NRT方程组外，还有其他族类的非线性薛定ding方程组，它们的动力学特性与de Broglie关系兼容。然而，值得注意的是，与时间相关的类似高斯波包解决方案的存在是NRT方程的一个独特特征，而上述非线性泛化方程族并没有共享。

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