Q. how to calculate radial nodes

A radial node is a radius \(r>0\) where the radial part of a hydrogenic wavefunction, \(R_{n,l}(r)\), is zero. To calculate the number of radial nodes for a given principal quantum number \(n\) and orbital quantum number \(l\), follow the steps below.

Step 1. Separate the hydrogenic wavefunction into radial and angular parts. The full stationary state is written as \(\Psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)\,Y_{l}^{m}(\theta,\phi)\). Radial nodes are solutions of \(R_{n,l}(r)=0\) with \(r>0\).

Step 2. Write the standard form of the hydrogenic radial function. Up to a normalization constant \(N_{n,l}\), the radial function has the form

\[
R_{n,l}(r)=N_{n,l}\,\left(\frac{2r}{n a_{0}}\right)^{l}\,L_{n-l-1}^{\,2l+1}\!\left(\frac{2r}{n a_{0}}\right)\,\exp\!\left(-\frac{r}{n a_{0}}\right).
\]

Here \(a_{0}\) is the Bohr radius, and \(L_{k}^{\alpha}(x)\) is an associated Laguerre polynomial of degree \(k\). The factors in this expression matter for counting zeros as follows.

Step 3. Identify which factors can produce zeros at \(r>0\). The exponential factor \(\exp\!\left(-\dfrac{r}{n a_{0}}\right)\) never vanishes for finite \(r\). The prefactor \(\left(\dfrac{2r}{n a_{0}}\right)^{l}\) vanishes only at \(r=0\). By convention the origin \(r=0\) is not counted as a radial node. The remaining factor that can produce positive, finite zeros is the associated Laguerre polynomial \(L_{n-l-1}^{\,2l+1}\left(\dfrac{2r}{n a_{0}}\right)\).

Step 4. Count the zeros of the Laguerre polynomial. An associated Laguerre polynomial \(L_{k}^{\alpha}(x)\) has degree \(k\), and therefore has \(k\) real positive roots (counting multiplicity) when \(\alpha>-1\). In our case \(k=n-l-1\) and \(\alpha=2l+1>-1\). Therefore the number of radial nodes equals the degree \(n-l-1\).

Step 5. State the result as a compact formula and note domain restrictions. The number of radial nodes is

\[
\text{radial nodes}=n-l-1,
\]

where \(n\) is a positive integer and \(l\) is an integer with \(0\le l\le n-1\). If the formula gives a negative number, interpret that as zero radial nodes, but for allowed quantum numbers the expression is nonnegative.

Step 6. Quick checks and examples. The total number of nodes (radial plus angular) for a hydrogenic orbital equals \(n-1\), and the number of angular nodes equals \(l\). Thus radial nodes \(=n-1-l\) as above. Examples: for a 1s orbital \(n=1\), \(l=0\), radial nodes \(=1-0-1=0\). For a 2s orbital \(n=2\), \(l=0\), radial nodes \(=2-0-1=1\). For a 2p orbital \(n=2\), \(l=1\), radial nodes \(=2-1-1=0\). For a 3p orbital \(n=3\), \(l=1\), radial nodes \(=3-1-1=1\).