The wave function itself, a complex function with positive and negative values, doesn't tell us
much about the structure of the atom or any connectivity it may have with other atoms. The complex
square of the wave function represents the *probability density of finding the electron* at a given
point in space when one looks (*i.e.* does an experiment). It does not say anything about where the
electron actually *is* at any moment, the solutions of the Schrödinger equation only
refer to which states are observable. The act of measuring throws the system into one of
the states that are solutions of the Schrödinger equation.

A recurrent idea of statistics is that temporal averages are equivalent to ensemble averages, *i.e.*
look for a while at one object moving randomly and you will see the same set of orientations that you
would see when taking a snapshot of many identical objects moving randomly. In the same way, we can
interpret the probability density as a representative of the electron density in an atom, molecule, or
solid. This makes the wave function useful to predict chemical bonding between atoms.

The radial solutions of the Schrödinger equation of the hydrogen atom, *R(r)*, are plotted
on the right. Each time the quantum number *n* increases, an additional node is created. At
*n*=1, the radial function is all positive. Its maximum is at *r*=0, *i.e.* the point in
space with the highest probability density of finding the electron is actually *inside* the nucleus!
That is why the term probability *density* is used: As we move outward along the radius, the volume
of a shell of equal thickness is getting larger and larger, thereby spreading out the probability over
a larger volume.

Each time the quantum number *l* is increased, one of the spherical nodes disappears again. It is
replaced by a planar node that goes through the nucleus. Therefore, only *l*=0 electrons have a
finite probability density at the nucleus.

If anyone is interested, here's the C code generating the images.

The diagram on the right shows cross sections of the full wave function *ψ(r,θ,φ)* in the polar
(*r*-*θ*) plane. This representation highlights the transformation of spherical nodes into
planar nodes as *l* increases.

It is also apparent that the wave function is spreading out into space as *n* increases, *i.e.*
that electrons with a small *n* are, on balance, nearer the nucleus. Given that the energy eigenvalue
increases with *n*, it matches the semi-classical expectation that electrons have a lower energy if
they are deep down in the Coulomb potential.

One more observation: As *l* increases, the additional planar nodes cause the wave function to
become less and less symmetric. This is compensated by the increasing number of equivalent states having
the same *n* and *l* but different *m*. (In the diagram, only *m*=0 states are shown.)
All (2*l*+1) states with the same value of *n* and *l* together form a perfectly spherical
distribution of probability density.

Note that since the probability density is the square of the wave function, it doesn't make any difference if the wave function is positive or negative.

l= |
0 | 1 | 2 | 3 | 4... |

letter: | s | p | d | f | g (then continue alphabetically) |

Instead of describing a state by listing the values of its quantum numbers, a common practice is to refer
to them by main quantum number, *n*, followed by a letter representing the value of *l* as shown
in the table.

Thus, a 2p state is one with *n*=2 and *l*=1 (the *m*=-1,0,+1 cases are sometimes
distinguished as 2p_{x}, 2p_{y}, and 2p_{z}); a 3d state is one with *n*=3 and
*l*=2.

The hydrogen-like atom consists of only two particles, a nucleus and one electron. In a two-body
system, the distance of each particle from the centre of gravity is constant. As the two particles
move, they orbit around the common centre of gravity, but their distance from the centre does not change.
In a reference frame tied to the larger mass (*i.e.* the nucleus - see Fig.), the centre of gravity
(green cross) and the smaller mass move on concentric circles (blue trajectories).

In a three-body system, on the other hand, the centre of gravity isn't generally located on the connecting line between any two of the particles. Its position changes as the particles change from a linear to a triangular arrangement. In the reference frame centred on the larger mass, this leads to an irregular motion of the centre of gravity which does not resemble the motion of the trajectories of any of the other particles.

This means that the reduced mass approach used when setting up the
Schrödinger equation is no longer accurate. While not exact solutions, the hydrogen
Schrödinger solutions can still be used as *approximations* for atoms with more electrons.
One consequence of this is that the energy eigenvalues become dependent
on the quantum numbers *l* and *m* as well as *n*, resulting in fine structure of the
spectra observed.

One way to adapt the hydrogen solutions to higher atoms is to take account of fully occupied shells of
electronic states. Whenever a set of states with the same *n* and *l* is filled completely
with electrons, the probability density from the different *m* states within the set add up to
a near spherically symmetric distribution. Such symmetrically distributed electrons can be treated together
with the nucleus as an *atom core* with a reduced effective nuclear charge,
*Z _{eff}*.

There are empirical rules to extend this idea of *shielding* quantitatively to electron clouds with
lesser symmetry. For example, electrons in an *m*=2 state have a higher shielding factor with
respect to *m*=1 states orientated along the same axis than they have on *m*=1 states orientated
perpendicularly.