Well most of us have heard that you cannot take the square root of , however, you can easily define a new number, called , to be just that. In equations, . Seems like fun, but what does it mean for the real world? Well, everything, almost.
Remember the discussion about superpositions? There we said that if some object, like an atom, can have states of spin up and spin down, it could also be in a superposition of the two, that is in the state of “spin up and down together”, which we wrote as . Well nothing stops us from also considering superpositions such as , that involve imaginary numbers. This can be made even more complicated by considering complex numbers, mixtures of imaginary an real numbers.
And where do we come across this kind of thing? As it turns out, the correct wave function (that is, state) of the electron in the hydrogen atom is a complex object, having both real and imaginary parts. The probability of finding the electron around the hydrogen is given by the square (absolute value squared, more precisely) of the wave function. Since it is difficult to draw, we present the picture of the wave function squared.