8.1 The hydrogen atom  (Page 5/11)

The probability of finding the electron in the region r to $r+dr$ (“at approximately r ”) is

Here P ( r ) is called the radial probability density function    (a probability per unit length). For an electron in the ground state of hydrogen, the probability of finding an electron in the region r to $r+dr$ is

where $a_0=0.5$ angstroms. The radial probability density function P ( r ) is plotted in [link] . The area under the curve between any two radial positions, say $r_1$ and $r_2$ , gives the probability of finding the electron in that radial range. To find the most probable radial position, we set the first derivative of this function to zero ( $dP\text{/}dr=0$ ) and solve for r . The most probable radial position is not equal to the average or expectation value of the radial position because $|{\psi }_{n00}{|}^2$ is not symmetrical about its peak value.

If the electron has orbital angular momentum ( $l\ne 0$ ), then the wave functions representing the electron depend on the angles $\theta$ and $$ $\varphi ;$ that is, ${\psi }_{nlm}=$ ${\psi }_{nlm}$ ( r , $\theta$ , $\varphi$ ). Atomic orbitals for three states with $n=2$ and $l=1$ are shown in [link] . An atomic orbital    is a region in space that encloses a certain percentage (usually 90%) of the electron probability. (Sometimes atomic orbitals are referred to as “clouds” of probability.) Notice that these distributions are pronounced in certain directions. This directionality is important to chemists when they analyze how atoms are bound together to form molecules.

A slightly different representation of the wave function is given in [link] . In this case, light and dark regions indicate locations of relatively high and low probability, respectively. In contrast to the Bohr model of the hydrogen atom, the electron does not move around the proton nucleus in a well-defined path. Indeed, the uncertainty principle makes it impossible to know how the electron gets from one place to another.

Summary

• A hydrogen atom can be described in terms of its wave function, probability density, total energy, and orbital angular momentum.
• The state of an electron in a hydrogen atom is specified by its quantum numbers ( n , l , m ).
• In contrast to the Bohr model of the atom, the Schrödinger model makes predictions based on probability statements.
• The quantum numbers of a hydrogen atom can be used to calculate important information about the atom.

Conceptual questions