The wave function \psi_{nlm} that describes a state of the electron in a hydrogen atom is called an atomic orbital. Despite the name, it is not a path the electron travels along; it is a function that tells where the electron is likely to be found. The probability of finding the electron in a small volume dV near the position \vec{r} is |\psi(\vec{r})|^2dV.
A common way to draw a probability distribution is to imagine measuring the position of the electron many times and placing a dot at each position found. The denser the dots, the more likely the electron is to be found there. The figure places dots one after another for the 1s orbital, on a plane through the nucleus, in proportion to the probability density.
The dots are denser closer to the nucleus. This is because the probability density of the 1s orbital is |\psi_{100}|^2 \propto e^{-2r/a_0}, largest at the nucleus and smaller farther away.
The peak at r = a_0 seen in the article on the hydrogen atom was a peak of the distribution obtained by multiplying the density by the area 4\pi r^2 of a sphere of radius r. Near the nucleus the density is large but the volume is small, so counting by distance, the most electrons are found around r = a_0. The peak of the density itself is at the nucleus.
A probability distribution has no sharp outer edge. When a shape is to be shown by a single surface, the boundary of the region in which the probability of finding the electron is 90% is often drawn. For the 1s orbital, 90% lies inside a sphere of radius 2.66a_0. The dotted circle in the figure is the cross section of this sphere.
An atomic orbital can be written as the product of a radial function and an angular function, \psi_{nlm} = R_{nl}(r)\,Y_{lm}(\theta, \varphi). The radial function decides the spread and the nodes that are spheres centered on the nucleus. The angular function decides how the orbital differs with direction and the nodes that are surfaces through the nucleus.
For an s orbital, with l = 0, the angular function Y_{00} is a constant. The probability distribution is the same in every direction, that is, spherically symmetric, and s orbitals differ from one another only in their radial functions.
The figure draws 1s, 2s and 3s at the same scale, and the larger n, the farther out they spread. The radius of the sphere containing 90% of the probability is 2.66a_0 for 1s, 9.13a_0 for 2s and 19.4a_0 for 3s. The average distances are 1.5a_0, 6a_0 and 13.5a_0, each equal to 1.5n^2a_0.
2s has one spherical surface at radius 2a_0 on which the probability is 0. 3s has two, at 1.90a_0 and 7.10a_0. Such a surface is called a node, and an s orbital has n - 1 nodes. The dotted circles in the figure are cross sections of the nodes.
The color of the dots shows the sign of the wave function. The wave function of 2s is proportional to (2 - r/a_0)e^{-r/2a_0}: positive inside the node and negative outside. The sign changes each time a node is crossed, so in 3s the regions go positive, negative, positive from the inside out.
The electron is never found on the surface of a node, but an electron in a single state is found both inside and outside the node. For an electron in 2s, the probability of being found inside the node is 5%, and outside it is 95%.
For a p orbital, with l = 1, the angular function changes with direction. The orbital with m = 0 is proportional to \cos\theta and has two lobes along the z axis. It is called the \mathrm{p}_z orbital. On the xy plane \cos\theta = 0, so the probability is 0, and this plane is a node.
The orbitals with m = 1 and m = -1 contain e^{i\varphi} and e^{-i\varphi}, and their probability distribution is a ring around the z axis. Adding and subtracting these two gives the \mathrm{p}_x orbital along the x axis and the \mathrm{p}_y orbital along the y axis, proportional to \sin\theta\cos\varphi and \sin\theta\sin\varphi. A sum of solutions with the same energy is also a solution with that energy.
The figure draws the 2p orbitals on the xz plane through the nucleus, with the node shown as a dotted line. The two lobes have different colors because the wave function has opposite signs there. The probability |\psi|^2 does not depend on the sign, but when the orbitals of two atoms overlap to form a molecule, the result depends on whether the overlapping parts have the same sign or opposite signs.
The three p orbitals have the same shape and differ only in direction. Adding their three probabilities gives a spherically symmetric result, because \cos^2\theta + \sin^2\theta\cos^2\varphi + \sin^2\theta\sin^2\varphi = 1.
The radial function of 2p is proportional to re^{-r/2a_0} and has no node. The distribution by distance is largest at r = 4a_0. For 3p, one spherical node is added to the radial function.
There are five d orbitals with l = 2, and as with the p orbitals, they are rearranged into real functions. Four of them, \mathrm{d}_{xy}, \mathrm{d}_{yz}, \mathrm{d}_{zx} and \mathrm{d}_{x^2-y^2}, have four lobes on a plane, with neighboring lobes of opposite sign.
\mathrm{d}_{zx} has its lobes on the xz plane, in the directions 45° between the axes. \mathrm{d}_{x^2-y^2} has its lobes on the xy plane, along the x and y axes. \mathrm{d}_{xy} is \mathrm{d}_{x^2-y^2} turned by 45° about the z axis.
The fifth, \mathrm{d}_{z^2}, is proportional to 3\cos^2\theta - 1 and consists of two lobes along the z axis and a ring around the xy plane. The ring has the opposite sign to the lobes. Its angular nodes are two cones making 54.7^\circ with the z axis.
The figure draws three of the 3d orbitals, each on a plane that shows its shape well. For \mathrm{d}_{zx}, the xy plane and the yz plane are nodes. The number of nodal surfaces through the nucleus equals l, which is two for a d orbital. The two cones of \mathrm{d}_{z^2} also count as two.
The colors and magnetic properties of compounds of transition metals such as iron and copper depend on the directions of the d orbitals. Depending on the directions of the surrounding atoms, the energies of the five d orbitals become slightly different.
An atomic orbital has two kinds of nodes: spheres on which the radial function is 0, and surfaces through the nucleus on which the angular function is 0. There are n - l - 1 spherical nodes and l nodes through the nucleus, n - 1 in all.
The figure shows the nodes of the orbitals up to n = 3 schematically. Dotted circles are spherical nodes, and solid lines are nodes through the nucleus. 3s, 3p and 3d, with the same n, all have two nodes, and only the breakdown differs.
In the one-dimensional square well too, the nth state had n - 1 nodes, and the more nodes, the higher the energy. In the hydrogen atom as well, states with the same number of nodes n - 1 have the same energy.
In atoms with two or more electrons, the shapes of the orbitals and the numbers of nodes are roughly the same as in hydrogen. For the same n, however, the energy comes to depend on l. Which orbitals the electrons fill first is taken up in the articles on the Pauli exclusion principle and the periodic table.
The directions of orbitals also bear on the shapes of molecules. Combining the 2s and the three 2p orbitals of a carbon atom gives four orbitals pointing to the vertices of a regular tetrahedron. This is why the four bonds of methane make angles of 109.5^\circ with each other.