Q.Describe the shape of the $d_{z^2}$ orbital and explain how it differs from the shape of the other four $d$ orbitals.

For a one-electron system, the Schrödinger equation's solution in spherical polar coordinates factorises as $\Psi(r,\theta,\varphi)=R(r)\cdot f(\theta)\cdot g(\varphi)$, letting the probability distribution be studied separately as a function of distance from the nucleus (the radial distribution) and as a function of direction (the angular distribution).

Radial distribution and nodes. Plotting the raw $R(r)^2$ against $r$ is misleading, since it ignores how much physical space exists at each distance; correctly weighting by the volume of a thin spherical shell, $dV=4\pi r^2\,dr$, gives the true radial probability curve, $4\pi r^2\cdot R(r)^2$ versus $r$. For hydrogen's 1s orbital this curve peaks exactly at the Bohr radius (0.529 Å), the electron's single most probable distance from the nucleus. For higher orbitals the curve can touch zero at one or more specific radii called radial nodes - spherical shells where the electron is essentially never found - and the general counting rule is

$\text{radial nodes}=n-l-1$ …