Hydrogen wavefunctions and orbital nodes

For the point-Coulomb Hamiltonian write $\psi_{n\ell m}=R_{n\ell}(r)Y_{\ell m}(\theta,\phi)$. Let $a=a_\mu/Z$ and $\rho=2r/(na)$. The radial factor is proportional to $\rho^\ell e^{-\rho/2}L_{n-\ell-1}^{2\ell+1}(\rho)$, with an associated Laguerre polynomial. Its positive roots produce spherical nodal surfaces. The origin factor for nonzero ℓ and asymptotic decay at infinity are not counted as positive-radius radial nodes.

The radial-node number is $n-\ell-1$. Increasing n at fixed ℓ adds a radial node, whereas increasing ℓ within one Coulomb shell removes one. This is a property of the separated Coulomb eigenfunctions, not a general rule that every three-dimensional state of a given energy has one fixed nodal geometry. Superpositions within a degenerate shell can change the shape and phase structure, and arbitrary superpositions need not retain one radial and one angular factor.

An s orbital has constant angular factor and no angular nodal surface. A real $p_z$ orbital has angular factor $\cos\theta$, so the plane z = 0 is a node. Real $p_x$ and $p_y$ orbitals are rotated combinations of the complex m eigenfunctions. Their differently oriented planes do not create different Coulomb energies. A lobe colour normally indicates wavefunction sign or phase; it does not indicate positive versus negative probability or opposite electric charges.

The familiar count of ℓ angular nodes is used for the conventional real separated orbital basis, with planes or cones counted in the usual way. It should not be interpreted as a basis-independent claim about every complex spherical harmonic's nodal surfaces. For example, $Y_{11}\propto\sin\theta e^{i\phi}$ has azimuthal phase winding rather than the nodal plane of a real p orbital. Specifying the basis is therefore essential when translating a quantum number into a particular three-dimensional picture.

For the normalized 1s orbital, spatial density decreases monotonically from the origin. Yet a thin shell at radius r has area proportional to $r^2$, and its probability per unit radius initially grows from zero. Combining the geometric factor with exponential decay produces a shell-density maximum at a nonzero radius. There is no contradiction: probability in a fixed tiny volume near the nucleus and probability in a whole spherical shell are different measurements.

For any normalizable continuous spatial density, the probability of one exact point is zero even if its density value is finite. An s orbital's finite density at the nucleus is therefore not a finite probability assigned to a point. Likewise, a radial node is a sphere where amplitude vanishes, while the sign change across it can affect interference with another state. Plot captions should distinguish amplitude, density per volume, and density per radius before a maximum or zero is interpreted.