---
title: Hydrogen Wave Functions and Orbitals
module: The Quantum Hydrogen Atom
moduleNumber: 2
lessonNumber: 2
order: 202
summary: >
  The hydrogen wave functions factor into a radial part Rₙℓ(r) and an angular
  spherical harmonic Yℓm(θ,φ). Squaring gives the probability cloud; the radial
  distribution P(r) = r²|ψ|² peaks at the Bohr radius for the ground state and at
  the Bohr orbits for excited states. The angular part fixes the s, p, and d
  orbital shapes that govern chemical bonding.
topics: [The Quantum Hydrogen Atom]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 7 — Atomic Physics; §7-3 The Hydrogen Atom Wave Functions"
draft: false
---

The [previous lesson](/atomic-physics/quantum-hydrogen-atom/schrodinger-3d-hydrogen)
extracted the quantum numbers and energies of hydrogen from the separated
Schrödinger equation. This lesson uses the wave functions themselves. Each one is
a product of a radial factor and an angular factor, and its square is a
probability density in three-dimensional space. The radial factor answers "how
far from the nucleus?"; the angular factor answers "in what direction?" Together
they replace the Bohr orbit with a diffuse cloud whose shape depends on $\ell$
and $m$.

## The complete wave function

Assembling the three separated factors, the hydrogen wave function is

$$
\psi_{n\ell m}(r,\theta,\phi) = C_{n\ell m}\,R_{n\ell}(r)\,f_{\ell m}(\theta)\,g_m(\phi)
= C_{n\ell m}\,R_{n\ell}(r)\,Y_{\ell m}(\theta,\phi),
$$

with $C_{n\ell m}$ fixed by normalization. The energy depends only on $n$, but
the wave function depends on all three quantum numbers, one from each coordinate.
For any $n$ there are $n$ values of $\ell$ and $2\ell + 1$ values of $m$, so most
energy levels carry several distinct wave functions — the degeneracy of the
inverse-square force.

The radial functions for the three lowest shells, in the hydrogenic form with
Bohr radius $a_0 = \hbar^2/\mu ke^2$, are:[^tl-73]

| $n$ | $\ell$ | $R_{n\ell}(r)$ |
| --- | --- | --- |
| $1$ | $0$ | $2\,a_0^{-3/2}\,e^{-r/a_0}$ |
| $2$ | $0$ | $\frac{1}{2\sqrt{2}}\,a_0^{-3/2}\left(2 - \frac{r}{a_0}\right)e^{-r/2a_0}$ |
| $2$ | $1$ | $\frac{1}{2\sqrt{6}}\,a_0^{-3/2}\,\frac{r}{a_0}\,e^{-r/2a_0}$ |
| $3$ | $0$ | $\propto \left(1 - \frac{2r}{3a_0} + \frac{2r^2}{27a_0^2}\right)e^{-r/3a_0}$ |
| $3$ | $1$ | $\propto \frac{r}{a_0}\left(1 - \frac{r}{6a_0}\right)e^{-r/3a_0}$ |
| $3$ | $2$ | $\propto \frac{r^2}{a_0^2}\,e^{-r/3a_0}$ |

Two structural facts read off the table. The exponential decay rate is set by
$n$ alone through $e^{-r/na_0}$, so higher shells extend farther. Near the
origin $R_{n\ell} \propto r^{\ell}$, so only $\ell = 0$ states are nonzero at the
nucleus; the higher the $\ell$, the more the centrifugal barrier pushes the
electron out.

## Probability density and the radial distribution

Born's rule makes $|\psi|^2\,\d\tau$ the probability of finding the electron in
the volume element $\d\tau$. In spherical coordinates the volume element is

$$
\d\tau = r^2\sin\theta\,\d r\,\d\theta\,\d\phi.
$$

Two different questions have two different answers. The **probability density**
$|\psi|^2$ is the chance per unit volume at a point. The **radial probability**
$P(r)\,\d r$ is the chance of finding the electron anywhere in the thin spherical
shell between $r$ and $r + \d r$, obtained by multiplying the density by the shell
volume $4\pi r^2\,\d r$ (for a spherically symmetric state):

$$
P(r) = 4\pi r^2\,|\psi|^2 \propto r^2\,|R_{n\ell}(r)|^2.
$$

$$
% caption: The radial probability weights the density by the shell volume 4πr²dr,
% which vanishes at the origin and grows outward.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% nucleus
\fill[black!70] (0,0) circle (1.6pt);
\node[black!70, anchor=north east] at (0,0) {nucleus};
% shell (two concentric circles, annulus)
\draw[acc, thick] (0,0) circle (2.3);
\draw[acc, thick] (0,0) circle (2.65);
\begin{scope}
  \clip (0,0) circle (2.65);
  \fill[acc!12] (0,0) circle (2.65);
  \fill[white] (0,0) circle (2.3);
\end{scope}
\draw[black, ->] (0,0) -- (2.3,0);
\node[black!70, anchor=north] at (1.15,-0.05) {$r$};
\draw[black, <->] ({2.3*cos(35)},{2.3*sin(35)}) -- ({2.65*cos(35)},{2.65*sin(35)});
\node[acc, anchor=west] at ({2.65*cos(35)},{2.65*sin(35)}) {$dr$};
\node[acc, anchor=west] at (1.9,1.9) {spherical shell};
\end{tikzpicture}
$$

### The ground state

For $n = 1$, both $\ell$ and $m$ are zero, the Laguerre polynomial is $1$, and

$$
\psi_{100} = C_{100}\,e^{-Zr/a_0},
\qquad
C_{100} = \frac{1}{\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}.
$$

The density $|\psi_{100}|^2 \propto e^{-2Zr/a_0}$ is maximal at the origin and
falls monotonically. The radial distribution multiplies this by $r^2$, which
vanishes at the origin, so $P(r)$ starts at zero, rises, and peaks. Setting
$\d P/\d r = 0$ gives the **most probable radius**

$$
r_{\text{mp}} = \frac{a_0}{Z},
$$

the Bohr radius for hydrogen.[^tl-73] The two curves say complementary things:
the electron is most _densely_ found at the nucleus but most _likely_ found in
the shell at $a_0$, because there is far more shell area out there.

$$
% caption: For the ground state, |ψ|² peaks at the nucleus while the radial
% probability P(r) = r²|ψ|² peaks at the Bohr radius a₀.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (5.6,0) node[right, black!70] {$r$};
\draw[->, black] (0,0) -- (0,3.2) node[above, black!70] {value};
% |psi|^2 : exponential decay from max at 0
\draw[acc, very thick] (0,3.0) .. controls (0.6,1.6) and (1.4,0.55) .. (2.6,0.22)
  .. controls (3.6,0.08) and (4.6,0.03) .. (5.3,0.02);
\node[acc, anchor=west] at (0.55,2.55) {density};
% P(r): rises then falls, peak near a0 (x=1.2)
\draw[black, very thick, dashed] (0,0) .. controls (0.6,1.8) and (1.0,2.5) .. (1.4,2.5)
  .. controls (1.9,2.5) and (2.6,1.4) .. (3.4,0.65)
  .. controls (4.2,0.28) and (4.9,0.12) .. (5.3,0.07);
\node[black, anchor=south] at (1.9,2.5) {$P(r)$};
% mark a0
\draw[black, dashed] (1.35,0) -- (1.35,2.5);
\node[black!70, anchor=north] at (1.35,0) {$a_0$};
\end{tikzpicture}
$$

### Excited states

For $n = 2$ the possibilities are $\ell = 0$ (the 2s state, spherically
symmetric) and $\ell = 1$ (the three 2p states):[^tl-73]

$$
\psi_{200} = C_{200}\left(2 - \frac{Zr}{a_0}\right)e^{-Zr/2a_0},
\qquad
\psi_{210} = C_{210}\,\frac{Zr}{a_0}\,e^{-Zr/2a_0}\cos\theta,
$$

with $\psi_{21,\pm 1} \propto (Zr/a_0)\,e^{-Zr/2a_0}\sin\theta\,e^{\pm i\phi}$.
The 2p radial distribution peaks at the second Bohr orbit,
$r_{\text{max}} = 2^2 a_0 = 4a_0$. The 2s distribution has two maxima: a large
one near $4a_0$ and a small subsidiary bump close to the nucleus, the signature
of the extra factor near the origin. This inner bump lets an $\ell = 0$ electron
penetrate the core of a heavier atom, which is decisive for the
[ordering of subshells](/atomic-physics/many-electron-atoms/periodic-table-atomic-spectra).

$$
% caption: Radial distributions for 1s, 2s, and 2p. The 2p peak sits at the
% second Bohr orbit 4a₀; the 2s curve adds a small inner bump near the nucleus.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.0,0) node[right, black!70] {$r$};
\draw[->, black] (0,0) -- (0,3.0) node[above, black!70] {$P(r)$};
% 1s peak near x=0.9
\draw[black, very thick] (0,0) .. controls (0.4,1.9) and (0.7,2.7) .. (0.95,2.7)
  .. controls (1.3,2.7) and (1.8,1.1) .. (2.4,0.45)
  .. controls (3.0,0.16) and (3.6,0.05) .. (4.0,0.03);
\node[black, anchor=south] at (0.95,2.72) {1s};
% 2p peak near x=3.6
\draw[acc, very thick, dashed] (0,0) .. controls (1.4,0.15) and (2.6,1.6) .. (3.5,1.75)
  .. controls (4.3,1.75) and (5.2,0.6) .. (5.8,0.28);
\node[acc, anchor=south] at (3.5,1.77) {2p};
% 2s: small bump near 1, big near 4
\draw[black, thick, densely dotted] (0,0) .. controls (0.5,0.85) and (0.9,0.9) .. (1.2,0.55)
  .. controls (1.6,0.2) and (2.0,0.12) .. (2.6,0.4)
  .. controls (3.4,0.85) and (4.0,1.25) .. (4.4,1.25)
  .. controls (5.0,1.25) and (5.5,0.6) .. (5.9,0.35);
\node[black, anchor=south] at (4.4,1.27) {2s};
\node[black, anchor=north] at (3.5,-0.05) {$4a_0$};
\draw[black, dashed] (3.5,0) -- (3.5,1.75);
\end{tikzpicture}
$$

## Radial nodes

The number of times $R_{n\ell}(r)$ crosses zero (for $r > 0$) is $n - \ell - 1$,
the **number of radial nodes**. The 1s and 2p functions have none; the 2s
function has one, at $r = 2a_0$, where the factor $(2 - r/a_0)$ vanishes. Nodes
carry no probability, and each one separates regions of opposite sign in the wave
function.

$$
% caption: The radial functions R₂₀ and R₂₁. R₂₀ (2s) crosses zero once at 2a₀;
% R₂₁ (2p) starts at zero, rises, and decays without a node.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.0,0) node[right, black!70] {$r$};
\draw[->, black] (0,-1.2) -- (0,2.4) node[above, black!70] {$R(r)$};
\draw[black] (0,0) -- (6.0,0);
% R20: positive near 0, crosses at 2a0 (x=2.2), negative dip, back toward 0
\draw[black, very thick] (0,2.0) .. controls (0.7,1.5) and (1.4,0.7) .. (2.2,0)
  .. controls (2.9,-0.6) and (3.6,-0.85) .. (4.3,-0.7)
  .. controls (5.0,-0.45) and (5.6,-0.2) .. (5.9,-0.1);
\node[black, anchor=west] at (0.15,2.05) {$R_{20}$ (2s)};
\fill[acc] (2.2,0) circle (1.8pt);
\node[acc, anchor=south] at (2.2,0.05) {node};
% R21: zero at origin, rises, single hump decays
\draw[black, very thick, dashed] (0,0) .. controls (0.9,1.0) and (1.6,1.55) .. (2.4,1.55)
  .. controls (3.3,1.55) and (4.4,0.7) .. (5.9,0.2);
\node[black, anchor=south] at (2.4,1.57) {$R_{21}$ (2p)};
\node[black, anchor=north] at (2.2,-0.05) {$2a_0$};
\end{tikzpicture}
$$

## Angular shapes and orbitals

The angular factor $|Y_{\ell m}(\theta,\phi)|^2$ fixes the directional
distribution, and it depends on $\ell$ and $m$ but never on the radial part.[^tl-73]

- **$\ell = 0$ (s).** $Y_{00}$ is constant, so the density is spherically
  symmetric — a round cloud.
- **$\ell = 1$, $m = 0$ (p).** The density is $\propto \cos^2\theta$, largest
  along the $z$ axis and zero in the equatorial plane: two lobes, a dumbbell.
- **$\ell = 1$, $m = \pm 1$.** The density is $\propto \sin^2\theta$, largest in
  the equatorial plane: a ring, or toroid, around the $z$ axis.
- **$\ell = 2$ (d).** Four-lobed cloverleaf patterns and their variants.

$$
% caption: Angular probability shapes: the s orbital is a sphere, the p orbital
% a two-lobed dumbbell, the d orbital a four-lobed cloverleaf.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% s: sphere
\draw[black, thick] (0,0) circle (0.85);
\node[black!70, anchor=north] at (0,-1.05) {s orbital};
% p: dumbbell (two vertical lobes)
\begin{scope}[xshift=3.2cm]
\draw[black, thick] (0,0.15) .. controls (0.75,0.55) and (0.75,1.35) .. (0,1.35)
  .. controls (-0.75,1.35) and (-0.75,0.55) .. (0,0.15);
\draw[black, thick] (0,-0.15) .. controls (0.75,-0.55) and (0.75,-1.35) .. (0,-1.35)
  .. controls (-0.75,-1.35) and (-0.75,-0.55) .. (0,-0.15);
\node[black!70, anchor=north] at (0,-1.55) {p orbital};
\end{scope}
% d: cloverleaf (four lobes)
\begin{scope}[xshift=6.6cm]
\foreach \a in {45,135,225,315}{
  \draw[black, thick] (0,0) .. controls ({1.15*cos(\a-16)},{1.15*sin(\a-16)}) and ({1.15*cos(\a+16)},{1.15*sin(\a+16)}) .. (0,0);
}
\node[black!70, anchor=north] at (0,-1.55) {d orbital};
\end{scope}
\end{tikzpicture}
$$

These angular shapes are the orbitals of chemistry. The directional lobes of the
p and d clouds determine the geometry of the bonds an atom can form.
A more literal picture plots the density as a dot cloud, dense where $|\psi|^2$
is large.

$$
% caption: A dot-density rendering of the ground-state cloud: each dot marks a
% sampled electron position, densest at the nucleus and thinning outward.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% concentric guide (faint)
\draw[black] (0,0) circle (2.2);
% scattered dots, denser near center
\foreach \x/\y in {0.05/0.1, -0.15/0.2, 0.2/-0.1, -0.1/-0.2, 0.3/0.25, -0.3/0.1, 0.1/0.35, -0.25/-0.3, 0.35/-0.25, 0.0/-0.4, 0.5/0.15, -0.45/0.3, 0.2/0.55, -0.2/0.5, 0.6/-0.4, -0.55/-0.35, 0.7/0.4, -0.7/0.2, 0.15/-0.7, -0.35/0.75, 0.85/-0.15, -0.8/-0.5, 0.4/0.9, -0.5/-0.85, 1.0/0.3, -1.05/0.1, 0.3/-1.1, -0.2/1.15, 1.2/-0.55, -1.15/-0.6, 0.9/0.95, -0.95/0.85, 1.45/0.2, -1.4/-0.25, 0.55/-1.45, -0.6/1.4, 1.7/-0.4, -1.65/0.5, 1.1/-1.3, -1.2/-1.25}{
  \fill[black] (\x,\y) circle (1.3pt);
}
\fill[black!75] (0,0) circle (1.6pt);
\node[black!70, anchor=north] at (0,-2.4) {ground-state density};
\end{tikzpicture}
$$

With the spatial wave functions in hand, the description is still incomplete: the
electron carries an intrinsic angular momentum the Schrödinger equation never
produced. That [spin](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession)
adds the fourth quantum number and splits the spectral lines.

[^tl-73]: **Tipler & Llewellyn**, §7-3 — the complete wave function $\psi_{n\ell m} = C_{n\ell m}R_{n\ell}Y_{\ell m}$, the ground-state normalization and most-probable radius $a_0/Z$, the $n=2$ radial distributions with the 2p peak at $4a_0$, and the angular ($s$/$p$/$d$) probability shapes.