Harmonic polynomials, hyperspherical harmonics, and atomic spectra
Harmonic polynomials, hyperspherical harmonics, and atomic spectra
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调和多项式、超球面谐波和原子光谱
DOI：
10.1016/j.cam.2009.02.057
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
J. Avery
中科院分区：
文献类型：
--
作者：
J. Avery
讨论了d维空间中单项式、齐次多项式和调和多项式的性质。的属性，导致公式的规范分解的齐次多项式和公式的谐波投影。球谐函数、盖根鲍尔多项式和超球谐函数的许多重要性质都是由这些公式导出的。调和投影也提供了处理角动量和广义角动量的替代方法。用这种方法可以导出角积分和超角积分的几个强有力的定理。这些纯粹的数学考虑有重要的物理应用，因为超球谐函数通过福克投影与库仑-斯特米安有关，而且斯特米安和广义斯特米安都在原子和分子的量子理论中表现出极其有用的作用。
The properties of monomials, homogeneous polynomials and harmonic polynomials in d-dimensional spaces are discussed. The properties are shown to lead to formulas for the canonical decomposition of homogeneous polynomials and formulas for harmonic projection. Many important properties of spherical harmonics, Gegenbauer polynomials and hyperspherical harmonics follow from these formulas. Harmonic projection also provides alternative ways of treating angular momentum and generalised angular momentum. Several powerful theorems for angular integration and hyperangular integration can be derived in this way. These purely mathematical considerations have important physical applications because hyperspherical harmonics are related to Coulomb Sturmians through the Fock projection, and because both Sturmians and generalised Sturmians have shown themselves to be extremely useful in the quantum theory of atoms and molecules.