Amplitudes are more than just tools for calculating probabilities: they tell us something about physical reality. When we deal with probabilities, we may think about them as numbers that quantify our lack of knowledge. Indeed, when we say that a particle goes through the upper or the lower slit with some respective probabilities, it does go through one of the two slits, we just do not know which one. In contrast, according to quantum theory, a particle that goes through the upper and the lower slit with certain amplitudes does explore both of the two paths, not just one of them. This is a statement about a real physical situation — about something that is out there and something we can experiment with.
The assumption that the particle goes through one of the two slits, but just that we do not know which one, is inconsistent with many experimental observations.
We have to accept that, apart from some easy to visualise states, known as the basis states, (such as the particle at the upper slit or the particle at the lower slit), there are infinitely many other states, all of them equally real, in which the particle is in a superposition of the two basis states. This rather bizarre picture of reality is the best we have at the moment, and it works, at least for now.
Physicists write such superposition states as11
Dirac notation will likely be familiar to physicists, but may look odd to mathematicians or computer scientists. Love it or hate it (and I suggest the former), the notation is so common that you simply have no choice but to learn it, especially if you want to study anything related to quantum theory.↩︎