Great question. As far as I know, there is no direct way to measure the electronic configuration (wavefunction) of a single atom. Basically, the electronic configurations are calculated from first principles, and then compared with a wide variety of experimental results to check their accuracy.


One such experimental result, is that atoms in a dilute gas, will absorb and emit single photons of light when their electrons move from one energy level (configuration) to another. Because the energy levels that these electrons occupy are quantized (only have certain values), the light is absorbed or emited only at certain precise frequencies. By comparing an atom's predicted energy levels to the way that a dilute gass of atoms interacts with light, one can tell if the predicted energy levels make sense.


How are the electronic configurations calculated? Basically, you must solve the Schodinger equation for each electron. This calculation is not possible analytically, but it can be solved numerically given certain simplifying assumptions.


This answer is pretty technical, so feel free to ask for clarification.
You might want to look on the web as well. A Google search on "electronic configuration atoms" will give 14,000 hits, which should keep you busy for a while.

Electron configurations for many metals are strongly dependent on the chemical and physical properties of the structure they are embedded in. It is exceedingly difficult to directly observe the electron structure of a single atom. However, when periodically appearing in a crystal lattice, is it often possible to measure properties generic to all atoms located at a particular site within the lattice. The techniques that have been used most offen are: Xray-crystallography, and Solid-Phase NMR. In crystallography, the scattering of Xrays is measured for a lattice of atoms in a sample at some realtively narrow frequency of x-rays. This scattering is off electrons in the crystal and the scattering angle and intensity is a function of the periodicity of the lattice to the wavelength of the Xrays. (A very similar effect is seen looking at the surface of a DVD in a bright light -- the color zones show the presence of frequency selective scattering.
Xrays, having much shorter wavelengths, preferentially scatter off atoms (or more precisely atomic or molecular bonding orbitals). It might seem that very little information can be gained this way -- however, such scattering experiments have been and remain the key technique to unlock and verify chemical structures. Solid Phase NMR is a more recent tool which can also help determine structure not by measurement of position -- but of measurement of the relative strength of the electrical and magnetic fields surrounding an atomic nucleus. In this case, the sample is probed with a radio frequency field in the presence of a strong magnetic field. Precession of a nuclear spins can be modulated and thus effect such fields either as absorption or emission of RF energy. This effect is directly effect by the local fields of the electron chemcial bonds. So you can determine how many different positions (in field strength) exist in the material. By using oriented sensos, one can measure such levels as the physical orientation is altered -- making a map which can be used to induce the structure.
Finally, for certain materials with stable surface properties, recently one can directly measure the orbital fields using atomic force microscopy. This is a really nifty technique for surface phenomena. Digital Instruments in Santa Barbara manufactures such microscopes. They have a fun and informative website:
click here