There are many ways to visualize the wavefunctions ("orbitals") of the hydrogen atom. The text discusses and shows several ways in Section 15.7 (see in particular Figures 15.23, 15.24, and 15.26), but here we consider another way.

To picture the overall size, shape, and orientation of the atom, imagine the following. You make repeated measurements of the instantaneous position of the electron relative to the nucleus, localizing the electron in some small volume. To keep track of these measurements, you construct a 3-D grid of transparent cubes, each cube representing a small volume in space. Whenever you make a measurement, you place a marker of some kind in the corresponding cube. After many such measurements, you stand back, look at the grid, and use the markers to picture the atom as best as it can be pictured. The pictures here were drawn by computer using just that analogy plus the exact H atom wavefunctions (or more correctly, the square of the wavefunctions, since it is the square that masures probability).

Each picture spans a radial distance of 30 times the Bohr radius (30a₀, or about 16 Å).

For some wavefunctions, a single view tells the whole story. These are the l = 0 (or s) wavefunctions, which are spherically symmetric, and the l = 1, m = 0 (or p_z) and l = 2, m = 0 (or d_z2) wavefunctions, which are cylindrically symmetric about the z axis. (In fact, all m = 0 wavefunctions have this symmetry, but only those for l = 1 and l = 2 are shown here.)

For other wavefunctions, we need three views, one along each of the axes, x, y, and z.

As you look through these pictures, take time to visualize the radial (or spherical) nodes and the angular (or planar) nodes. Recall that a wavefunction with principal quantum number n has n – 1 total nodes, and that the l quantum number equals the number of planar nodes, leaving n – l – 1 spherical nodes.

These pictures were generated by a computer program that simulated 50,000 measurements of the electron position.