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The Radial-Angular Decomposition

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2000)

Abstract

Symmetry plays an important role in quantum mechanics. Closed solutions of quantum mechanical problems are mostly determined with help of symmetry properties of the underlying Schrödinger equation. This holds particularly for one-electron problems with rotationally symmetric potentials. In this case the solutions split into products of problem-dependent radial parts and angular parts that are built up from three-dimensional spherical harmonics. The most prominent example is the Schrödinger equation for hydrogen-like atoms. The knowledge about its solutions is basic for our understanding of chemistry. The solutions of the Schrödinger equation for a general system of N electrons moving in the field of a given number of clamped nuclei unfortunately do not attain such a simple form. The norms that we introduced to measure their mixed derivatives are however invariant to rotations of the coordinates of the single electrons.We therefore decompose the solutions of the N-particle equation in this chapter into tensor products of three-dimensional angular momentum eigenfunctions, the decomposition that reflects this rotational invariance.

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Correspondence to Harry Yserentant .

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© 2010 Springer Berlin Heidelberg

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Yserentant, H. (2010). The Radial-Angular Decomposition. In: Regularity and Approximability of Electronic Wave Functions. Lecture Notes in Mathematics(), vol 2000. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12248-4_9

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