Hydrogenic Radial Wave Function

The solution to the hydrogenic Schrödinger equation can be written as the product of the radial R and angular Y wave functions:

ψnlm = R nl(r) Y l m(θ,ϕ)

The integers n, l, and m are known as the principal, angular momentum, and angular momentum projection quantum numbers. Note the principal quantum number determines the wave function energy and that all states with the same n value have the same binding (i.e., negative) energy. The ground state, n=1, has a binding energy E0 of 13.4 ev and this binding energy decreases as 1/n2 as shown in the simulation.

This simulations displays the radial wave functions

R nl(r) = A nl er/na0 [(r/na0) l+1/r] v n(r/na0) ,

where A nl = [(2/(na0)3(nl − 1)!/(2n[(n + l)!]3)] is the normalization for the radial wave function. In addition, v n( r/na0) = L 2l+1 n−l−1(2r/na0) are the associated Laguerre polynomials. The unnormalized radial wave functions, above without A nl, are shown in this simulation. In the animation, distances are given in terms of Bohr radii, a0. Enter values of n and l to see the radial wave function that results. The principal quantum number n can take any value starting at 1 and the value value of l can be 0, 1, 2, ... n−1. The value of m can be -l, -l+1, ... , -2, -1, 0, 1, 2, ... , l-1, l. Thus, there are n2 radial wave functions for each n value.

References:

Credits:

The EJS JavaScript version of this simulation was developed by Wolfgang Christian from the original Java version. This Java is also available in the ComPADRE OSP Collection.
Information about EJS is available at: <http://www.um.es/fem/Ejs/>.