41.2 The Hydrogen Atom: Wave Functions and Probabilities

You learned in Chapter 40 that the probability of finding a particle in a small interval of width $δ x$ at the position $x$ is given by

Prob(in δ x at x) = | ψ (x) |^{2} δ x = P (x) δ x

where $P (x) = | ψ (x) |^{2}$ is the probability density. This interpretation of $| ψ (x) |^{2}$ as a probability density lies at the heart of quantum mechanics. However, $P (x)$ was for a one-dimensional wave function.

A photograph of a nebula pattern with red colored dust enclosing a star in the center. — The red color of this nebula is due to the emission of light from hydrogen atoms. The atoms are excited by intense ultraviolet light from the star in the center. They then emit red light $(λ = 656 nm)$ in a $3 \to 2$ transition, part of the Balmer series of spectral lines emitted by hydrogen.

For a three-dimensional atom, the wave function is $ψ (x, y, z)$ , a function of three variables. We now want to consider the probability of finding a particle in a small volume of space $δ V$ at the position described by the three coordinates $(x, y, z)$ . This probability is

Prob(in δ V at x, y, z) = | ψ (x, y, z) |^{2} δ V

(41.6)

We can still interpret the square of the wave function as a probability density.

In one-dimensional quantum mechanics we could simply graph $P (x)$ versus $x$ . Portraying the probability density of a three-dimensional wave function is more of a challenge. One way to do so, shown in FIGURE 41.4, is to use denser shading to indicate regions of larger probability density. That is, the amplitude of $ψ$ is larger and the electron is more likely to be found in regions where the shading is darker. These figures show the probability densities of the $1 s$ , $2 s$ , and $2 p$ states of hydrogen. As you can see, the probability density in three dimensions creates what is often called an electron cloud around the nucleus.

The figure consists of four parts that illustrate the probability densities of the electron in the 1s, 2s, and 2p states of hydrogen using electron cloud.

FIGURE 41.4 Full Alternative Text

These figures contain a lot of information. For example, notice how the $p$ electrons have directional properties. These directional properties allow $p$ electrons to “reach out” toward nearby atoms, forming molecular bonds. The quantum mechanics of bonding goes beyond what we can study in this text, but the electron-cloud pictures of the $p$ electrons begin to suggest how bonds could form.