Wavefunctions and quantum numbers are the foundations of our current understanding of atomic structure, but this is advanced stuff, so don’t try and jump in without having a solid understanding of atomic orbitals and sub-orbitals and electron configurations.
First off, we need to realise that the electron, like all matter, has both wave and particle properties – we are very definitely working in the quantum realm now.
In 1924 Louis de Broglie suggested that for any object, the wavelength is inversely proportional to its mass and velocity:
For everyday objects, their wavelength is so very tiny (because of the very tiny value of Planck’s constant) that is is completely inconsequential – we don’t notice that buses and cats and pencils have wave properties. But this is not the case for electrons.
In the 1920s, whilst thrashing through the implications of quantum mechanics, Werner Heisenberg discovered a problem in the way that the basic physical properties of a particle with significant wave properties, like electrons, could be measured. We cannot know their precise position and momentum simultaneously – the Heisenberg Uncertainty Principle. So, if we cannot know the precise location of an electron in an atom, it would be more accurate to consider the probability of an electron being in a certain volume of space.
Erwin Schrödinger incorporated these ideas into a mathematical model, introducing us to the concept of the wavefunction.
- The wavefunction for an electron is calculated by solving the Schrödinger equation and contains detailed information on the behaviour of the electron. Unfortunately this is in a totally abstract way which is not so useful and the equation can only be solved for the hydrogen atom, also not very useful.
- Max Born then suggested that the square of the wavefunction is proportional to the probability of finding an electron within a small volume of space.
- Electrons are spread throughout this volume of space (remember that we are now thinking of electrons as waves, not particles) rather than fixed points or in fixed orbits (the Bohr model of the atom we used at GCSE). This volume of space is an atomic orbital.
When Schrödinger did manage to solve his equation for the hydrogen atom, it showed that the electron could only possess certain energies – these correspond exactly to the energy levels seen in atomic spectra.
- Each energy level is referred to by its principle quantum number, n, and it is an integer
n=1, n=2, n=3, n=4 etc.
Solving the equation also produces two other quantum numbers that help to describe the properties of a particular atomic orbital.
- The secondary quantum number (also known as the angular or orbital quantum number), l, which can be any whole number from 0 to (n-1). We only need to concern ourselves with values of l = 0, 1, 2 & 3 in practice.
- The magnetic quantum number, ml, which can be any value from -l to +l.
What does all this mean?
n gives us the energy of the orbital
l gives us the shape of the orbital (l=0 corresponds to an s orbital, l=1 corresponds to a p orbital, l=2 corresponds to a d orbital and l=3 corresponds to an f orbital)
ml gives us the orientation of the orbital in 3D space
1st energy level: n=1; l = 0 ; ml=0 so we can only have a 1s orbital
2nd energy level: n=2; l = 0 in which case ml=0 and we have a 2s orbital
l =1 (a 2p orbital) and ml= -1, 0, +1 (so that gives us three p sub-orbitals)
3rd energy level: n=3; l = 0 in which case ml=0 and we have a 3s orbital
l = 1 (a 3p orbital) and ml= -1, 0, +1 (so that gives us another three p sub-orbitals)
l = 2 (a 3d orbital) and ml= -2, -1, 0, +1, +2 (which gives us five d sub-orbitals)
The wavefunction also gives us information about the shape of an atomic orbital. The shape is shown by drawing a ‘boundary surface’ which contains about 95% of the wavefunction (remember this represents the probability of the electron being in this volume of space). 95% of the electron density will be found within the boundary surface.
As I mentioned earlier, Schrödinger’s equation can’t be solved for atoms other than hydrogen, but with a few assumptions, everything we’ve talked about so far holds true for atoms with more than one electron. However, there is one very important difference between the energy levels of a hydrogen atom compared with the energy levels of other atoms.
- For hydrogen, the energy of an orbital depends only on n, the principal quantum number. Taking the 2s and 2p orbitals as an example, since both belong to n=2, they both have the same energy. They are said to be ‘degenerate’.
- For other atoms, the energy of an orbital depends on l, the secondary quantum number. So, l=0 for a 2s orbital and l=1 for a 2p orbital. This means that the 2s orbital is of lower energy than the 2p orbital (but each of the three 2p orbitals is degenerate).
And just to complete the picture, there is a fourth quantum number, ms, which tells us the spin of the electron. An electron can be spinning clockwise (ms = +½) or anticlockwise (ms = -½).