Abstract
A general formula for angular integrations in many‐dimensional spaces (derived in a previous paper) is applied to several problems connected with solution of the Schrödinger equation for many‐particle systems. Matrix elements of the Hamiltonian are derived for cases where the potential can be expressed in terms of functions of the generalized radius multiplied by polynomials in the m coordinates. The theory of hyperspherical harmonics is reviewed, and a sum rule is derived relating the sum over all the harmonics belonging to a particular eigenvalue of angular momentum to the Gegenbauer polynomial corresponding to that eigenvalue. A formula is derived for projecting out the component of an arbitrary function corresponding to a particular eigenvalue of generalized angular momentum, and the formula is applied to polynomials in the m coordinates. An expansion is derived for expressing a many‐dimensional plane wave in terms of hyperspherical harmonics and functions which might be called “hyperspherical Bessel functions.” It is shown how this expansion may be used to calculate many‐dimensional Fourier transforms. A formula is derived expressing the effect of a group‐theoretical projection operator acting on a many‐dimensional plane wave. Finally, the techniques mentioned above are used to expand the Coulomb potential of a many‐particle system in terms of Gegenbauer polynomials.