弹性薄板的量子涨落和热涨落
Quantum and thermal fluctuations of a thin elastic plate
摘要：
我们考虑一个哈密顿描述的振动的固定，弹性圆板。这个系统的哈密顿量具有两个不同贡献的势能：一个取决于板的局部平均曲率，另一个取决于其高斯曲率。我们使用一个完整的，正交规范的本征函数的固定，振动板，这个模型。由此产生的量子是薄圆板的弯曲声子。作为一个应用，我们使用这种量子化的描述来计算任意温度下的板的位移波动。我们比较的波动轮廓与张力下的弹性膜。在低温下，我们发现，虽然这两个配置文件有一个圆环的局部最大值，环膜轮廓更加明显和尖锐。我们还注意到，随着温度的升高，板轮廓发展出两个额外的极值环。
英文摘要：
We consider a Hamiltonian description of the vibrations of a clamped, elastic circular plate. The Hamiltonian of this system features a potential energy with two distinct contributions: one that depends on the local mean curvature of the plate, and a second that depends on its Gaussian curvature. We quantize this model using a complete, orthonormal set of eigenfunctions for the clamped, vibrating plate. The resulting quanta are the flexural phonons of the thin circular plate. As an application, we use this quantized description to calculate the fluctuations in displacement of the plate for arbitrary temperature. We compare the fluctuation profile with that from an elastic membrane under tension. At low temperature, we find that while both profiles have a circular ring of local maxima, the ring in the membrane profile is much more pronounced and sharper. We also note that with increasing temperature the plate profile develops two additional rings of extrema.
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