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Acta Sinica Quantum Optica

The Angular Momentum Eigenfunctions in the Ultraspherical Space

Wei Zhi-yong

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The complete set of angular momentum eigenfunctions in the ultraspherical multidimensional space are obtained. The orthogonality relations, eigenvalues of the generalized angular momentum, expansion expression of the plane wave in the multidimensional space and other relations are given.

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Generalization to Three Dimensions

Walter Wilcox

Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. ƒe spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. ƒe angular momentum eigenvalues and eigenstates are determined using ladder operators and the connection to spherical harmonics is made. ƒe stage is set for understanding the Schrödinger radial equation.

CRC Press eBooks•Apr 19, 2016•0 citations
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.
Eigenfunctions and eigenvalues of the operator of the square of the angular momentum are studied. It is shown that neither from the requirement for the eigenfunctions be normalizable nor from the commutation relations it is possible to prove that the eigenvalues spectrum is a set of only integer numbers (in units ℏ=1\hbar=1ℏ=1). We present regular, normalizable eigenfunctions with the non-integer eigenvalues thus demonstrating that a non-integer angular momentum is admissible from the theoretical viewpoint.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. ƒe spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. ƒe angular momentum eigenvalues and eigenstates are determined using ladder operators and the connection to spherical harmonics is made. ƒe stage is set for understanding the Schrödinger radial equation.