Schrödinger’s wave equation:
The heart of quantum mechanics is the Schrödinger’s wave equation, a description of the wave properties of an electron. In Cartesian coordinates, one form of the wave equation looks like this:
Where ψ is the wave function, ħ = h/2π and h is Planck’s constant, m is mass of the electron, Ze2 is the nuclear charge, and E is energy.
The wave equation has a set of solutions called wave functions, ψ. For the hydrogen atom, these solutions are similar to the energies and orbits of the Bohr model. The function cannot be solved for an atom with many electrons or a molecule. We can find approximate solutions by setting boundary conditions on the function and by making simplifications, such as treating a bond as though it were a spring.
Although a set of wave functions describe the energy of an electron, they don’t really correspond to anything we can visualize. If we square the wave functions, however, we get a new set of functions that we can almost understand. The square of a wave function, ψ2, is known as the probability density. The probability density is a map of the volume around the nucleus likely to contain an electron, and, just as important, it gives us an idea of where the electron is most likely not going to be.
An orbital is a possible energy state of an electron. Most people find it easier to think visually, so they link the idea of an orbital with its probability density. In that respect, an orbital is a volume likely to contain an electron with a specific energy. For the simplest case of a single electron in a hydrogen atom at its lowest energy, the wave function and probability density match the Bohr model, except that we are stuck with the notion of an electron probably somewhere in a spherical volume instead of traveling in a circular orbit. At higher energies, the probability density volumes have more complicated shapes, and it becomes more difficult to distinguish among them.