Wave functions, measurements, and electron density

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.

Spherical and planar nodes

Fig.: The radial eigenfunctions of the hydrogen atom.

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.

Fig.: Polar cross sections of the eigenfunctions of the hydrogen atom.
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 (2l+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.

Spectroscopic notation

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 2px, 2py, and 2pz); a 3d state is one with n=3 and l=2.

Atoms with more than one electron

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.

Fig.: Shielding of the nuclear charge by inner electrons.

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, Zeff.

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.

Now we're ready to make molecules and solids.