School of Mathematical Sciences, University of Nottingham, , Nottingham NG7 2RD, UK.
Philos Trans A Math Phys Eng Sci. 2013 Dec 16;372(2007):20120521. doi: 10.1098/rsta.2012.0521. Print 2014 Jan 28.
We discuss the statistical properties of the volume of the nodal set of wave functions for two paradigmatic model systems which we consider in arbitrary dimension s≥2: the cuboid as a paradigm for a regular shape with separable wave functions and planar random waves as an established model for chaotic wave functions in irregular shapes. We give explicit results for the mean and variance of the nodal volume in the arbitrary dimension, and for their limiting distribution. For the mean nodal volume, we calculate the effect of the boundary of the cuboid where Dirichlet boundary conditions reduce the nodal volume compared with the bulk. Boundary effects for chaotic wave functions are calculated using random waves which satisfy a Dirichlet boundary condition on a hyperplane. We put forward several conjectures on what properties of cuboids generalize to general regular shapes with separable wave functions and what properties of random waves can be expected for general irregular shapes. These universal features clearly distinguish between the two cases.
我们讨论了两种典范模型系统中波函数节点集体积的统计性质，我们在任意维度 s≥2 下考虑这两种典范模型系统：长方体作为具有可分离波函数的规则形状的典范，以及平面随机波作为不规则形状中混沌波函数的既定模型。我们给出了任意维度下节点体积的均值和方差的显式结果，以及它们的极限分布。对于平均节点体积，我们计算了长方体边界的影响，其中狄利克雷边界条件与体积相比会减少节点体积。对于混沌波函数，我们使用满足超平面上狄利克雷边界条件的随机波来计算边界效应。我们提出了几个关于长方体的哪些性质可以推广到具有可分离波函数的一般规则形状，以及随机波的哪些性质可以预期到一般不规则形状的猜想。这些普遍特征清楚地区分了这两种情况。
