What is the Hydrogen Radial Wavefunction Calculator?
This tool computes the normalized radial wavefunction R(r) of a hydrogen-like (one-electron) atom — the part of the electron wavefunction that depends only on the distance r from the nucleus. It also returns r·R(r), since the radial probability density is proportional to (r·R(r))². It is pure quantum mechanics and applies universally. Distances are measured in Bohr radii (\(a = a_0 = 1\)).
How to use it
Choose the atomic number Z (Hydrogen Z=1 or the Helium ion He+ Z=2), enter the principal quantum number n (1, 2, 3, ...) and the azimuthal quantum number l (0 to n−1). Then set the starting radius, the step size, and the number of points to build a table you can plot.
The formula
With the dimensionless variable \(\rho = 2Zr/(na)\) and \(a = 1\), the normalized radial wavefunction is $$R_{n,l}(r) = N\,e^{-\rho/2}\,\rho^{\,l}\,L_{n-l-1}^{\,2l+1}(\rho),$$ where the normalization constant $$N = \sqrt{\left(\frac{2Z}{na}\right)^{3}\frac{(n-l-1)!}{2n\,(n+l)!}}.$$ The associated Laguerre polynomial uses $$L_p^q(x) = \sum (-1)^i \binom{p+q}{p-i} \frac{x^i}{i!}.$$
Worked example
For Z=1, n=2, l=0 (the 2s orbital): $$R_{2,0}(r) = \frac{1}{2\sqrt{2}}(2 - r)\,e^{-r/2}.$$ At r=0, \(R = \frac{1}{2.828427}\cdot 2 = 0.707107\). At r=2 there is a radial node where \(R=0\). This matches the calculator output.
FAQ
Why is there sometimes a leading minus sign? The overall sign is a phase convention with no physical meaning; (r·R)² is sign independent.
Why is R(0)=0 for l≥1? Because \(\rho^{\,l} = 0\) at r=0 when \(l\ge 1\).
What units does R(r) have? \(a_0^{-3/2}\), since the wavefunction is normalized over three-dimensional space measured in Bohr radii.
