So what does it look like? The Schrödinger equation for single-electron
Coulomb systems in spherical coordinates is

This type of equation is an example of a

Obviously, we are not going to go through the solution of the Schrödinger
equation, but we can understand something about its mechanics and
the solutions from a few simple considerations. Remember that the
Schrödinger equation is set up starting from the classical energy, which we
said takes the form

which we can write as

where

The term is actually dependent only on and , so it is purely angular. Given the separability of the energy into radial and angular terms, the wave function can be decomposed into a product of the form

Solution of the angular part for the function yields the allowed values of the angular momentum and the -component . The functions are then characterized by the integers and , and are denoted . They are known as

for , there is just one value of , , and, therefore,
one spherical harmonic, which turns out to be a simple constant:

For , there are three values of , , and, therefore, three functions . These are given by

Remember that

For , there are five values of , , and, therefore, five spherical harmonics, given by

The remaining function is characterized by the integers
and , as this function satisfies the radial part
of the Schrödinger equation, also known as the *radial Schrödinger equation*:

Note that, while the functions are not particular to the potential , the radial functions are particular for the Coulomb potential. It is the solution of the radial Schrödinger equation that leads to the allowed energy levels. The boundary conditions that lead to the quantized energies are and . The radial parts of the wave functions that emerge are given by (for the first few values of and ):

where is the Bohr radius

The full wave functions are then composed of products of the radial and angular parts as

At this points, several comments are in order. First, the
integers that characterize each state are known as
the *quantum numbers* of the system. Each of them corresponds
to a quantity that is classically conserved. The number
is known as the *principal* quantum number, the
number is known as the *angular* quantum number,
and the number is known as the *magnetic* quantum
number.

As with any quantum system, the wave functions
give the probability amplitude for finding the electron
in a particular region of space, and these amplitudes are used
to compute actual probabilities associated with measurements
of the electron's position. The probability of finding the
electron in a small volume element of space around the
point
is

What is ? In Cartesian coordinates, is the volume of a small box of dimensions , , and in the , , and directions. That is,

In spherical coordinates, the volume element is a small element of a spherical volume and is given by

which is derivable from the transformation equations.

If we integrate over a sphere of radius , we should obtain
the volume of the sphere:

which is the formula for the volume of a sphere of radius .

**Example**: The electron in a hydrogen atom () is in the
state with quantum numbers , and . What is
the probability that a measurement of the electron's
position will yield a value ?

The wave function
is

Therefore, the probability we seek is

Let . Then

After integrating by parts, we find

which is relatively large given that this is at least two Bohr radii away from the nucleus!

The part of the probability involving the product

is known as the

Another point concerns the number of allowed states for each
allowed energy. Remember that each wave function corresponds to
a probability distribution in which the electron can be found
for each energy. The more possible states there are, the more varied
the electronic properties and behavior of the system will be.

For , there is one energy and only one wave function
.

For , there is one energy and four possible
states, corresponding to the following allowable values
of and

Thus, there are four wave functions , , , and . Whenever there is more than one wave function corresponding to a given energy level, then that energy level is said to be