## 1.3 Superpositions

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, classically, when we say that “a particle goes through the upper or the lower slit with some respective probabilities”, what we really mean is that it does go through one of the two slits, but we just do not know which one for sure. 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 with which we can experiment.

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 as22 |\psi\rangle=\alpha |\text{upper slit}\rangle +\beta |\text{lower slit}\rangle, meaning the particle goes through the upper slit with amplitude \alpha, and through the lower slit with amplitude \beta. Mathematically, you can think about this expression as a vector |\psi\rangle in a two-dimensional complex vector space written in terms of the two basis vectors |\text{upper slit}\rangle and |\text{lower slit}\rangle. You could also write this vector as a column vector with two complex entries \alpha and \beta, but then you would have to explain the physical meaning of the basis states. Here, we use the Dirac notation |\phantom{0}\rangle, introduced by Paul Dirac (1902–1984) in the early days of the quantum theory as a useful way to write and manipulate vectors. In Dirac notation you can put into the box |\phantom{0}\rangle anything that serves to specify what the vector is: it could be |\uparrow\rangle for spin up and |\downarrow\rangle for spin down (whatever this technical terminology “spin” means), or |0\rangle for a quantum bit holding logical 0 and |1\rangle for a quantum bit holding logical 1, etc. As we shall soon see, there is much more to this notation, and learning to manipulate it will help you greatly.

1. Dirac notation will likely be familiar to physicists, but may look odd to mathematicians or computer scientists. Love it or hate it (and we 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.↩︎