At atomic scales electrons cannot be thought of as points; instead they are smeared out probability distributions. They don't exist at any given point, there's a chance for a given electron to be found throughout a whole region of space, and the probability of finding it at any given point is given by a probability distribution. These probability distributions are called wave functions, and given an electron's wave function you can calculate the likelihood of getting different results when you take a measurement of the electron. It is a strange aspect of quantum mechanics that you can't calculate exactly what you will measure, you can only establish the probabilities of each possible outcome.
Another aspect of quantum mechanics is that if you measure, say, the energy of an electron in an atom, you can only get one of a certain set of discrete values, and never any energy in between those values. The energy of the electron is quantized
. In general, if you measure an electron's energy you have a certain probability to get a result corresponding to the first energy level, a probability to find it in the second energy level, and so on. This is also the case for some other things you can measure, like angular momentum.
However, there are certain wave functions that correspond to exactly one value of energy; that is, if you have an electron with this wave function, you are guaranteed to get a certain energy value when you measure it. In fact, there is a special set of wave functions with the following three properties:
- They each have a definite energy level.
- They each have a definite total angular momentum around the nucleus.
- They each have a definite angular momentum around the z axis.
These wave functions are the atomic orbitals that are so important in chemistry. If you calculate the shapes of the wave functions that satisfy these properties, you get the shapes shown on the Wikipedia page. They are listed in a table indexed by the variables n, l, and m. n corresponds to the energy level, l corresponds to the total angular momentum, and m corresponds to the angular momentum around the z axis. For example, you can see that orbitals with high m (angular momentum around the z axis), like the ones on the very right of the Wikipedia table, are sort of flattened out by the centrifugal force from spinning fast around a vertical axis.