The s-orbitals only have a radial part.

The principal quantum number arose in the solution of the radial part of the wave equation as shown below.

This operator appears also in the Schrödinger equation of the hydrogen atom after the radial part is separated off.

The corresponding ordinary differential operator is the radial part of the Laplacian operator on 2-dimensional hyperbolic space.

This wave function is expressed most naturally in spherical polar coordinates and so it has a radial part, and two angular parts.

STOs have the following radial part:

However, it is possible to solve the radial part of the integral so that only the integration over the azimuth angle remains to be done numerically.

The Hamilton-Jacobi equation gives an integral solution for the radial part S(r)

Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin.

It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every r.

In the case of a H-atom there are three such integers --one for the radial part, and one each for the two angular parts.

The database determines shapes of (atom-centered) basis functions (in fact radial part of these function) optimised for one or the other criteria.

Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers:

The radial parts of these atomic orbitals are sometimes numerical tables or are sometimes Slater orbitals.

The radial part of the solution varies from one potential to another, but the harmonics are always the same and are a consequence of spherical symmetry.

This equation arises in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom.

The radial part of the Laplacian in this case leads to a hypergeometric differential equation, the theory of which was treated in detail by Weyl.

A visualization of all common and uncommon atomic orbitals, from 1s to 7g (Note that the radial part of the expressions given corresponds to Slater orbitals rather than Gaussians.

The interpretation is that when a U(1) gauge field does not require quantized charges, it is possible to keep only the angular part of the Higgs oscillations, and discard the radial part.

These approximate methods were (and are) often used together with the central field approximation, to impose that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction.

Laplace's equation in spherical coordinates, such as are used for mapping the sky, can be simplified, using the method of separation of variables into a radial part, depending solely on distance from the centre point, and an angular or spherical part.