The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to a maximum value. This quantity and its derivative, the density of states, have important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, Bose-Einstein condensation and the Casimir effect. Weyl's theorem states that, in the limit of large wavenumbers, the cumulative state number depends only on the volume of the bounding domain and not on its shape. Corrections to this behavior are well known and depend on the surface area of the bounding domain, its curvature and other features. We describe several projects that allow readers to investigate this dependence for three bounding domains - a rectangular box, a sphere, and a circular cylinder. Quasi-one- and two-dimensional systems can be analyzed by considering various limits. The projects have applications in statistical mechanics, but can also be integrated into quantum mechanics, nuclear physics, or computational physics courses.

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外尔问题：一种计算方法

波动方程的特征值在有界域中的分布称为 Weyl 问题。我们描述了几个与累积状态数相关的计算项目，累积状态数定义为波数达到最大值的状态数。这个量及其导数，即态密度，在核物理、简并费米气体、黑体辐射、玻色-爱因斯坦凝聚和卡西米尔效应中有重要应用。外尔定理指出，在大波数的限制下，累积状态数仅取决于边界域的体积而不取决于其形状。对此行为的修正是众所周知的，并且取决于边界域的表面积、其曲率和其他特征。我们描述了几个项目，让读者可以研究三个边界域的这种依赖性 - 一个矩形框、一个球体和一个圆柱体。可以通过考虑各种限制来分析准一维和二维系统。这些项目在统计力学中有应用，但也可以集成到量子力学、核物理或计算物理课程中。