In classical physics, particles and waves are mutually exclusive.

• A particle is described by its position $\vec{r}(t)$ and momentum $\vec{p}(t)$, and always has a well-defined trajectory in space-time.
• A wave is an excitation of a field and is specified by its wavefunction, e.g. $\psi(\vec{r},t) = Ae^{i(\vec{k}\cdot\vec{r} - \omega t)}$. It is spread out over space at all times.

In quantum mechanics, the same physical system, e.g. an electron or light, sometimes displays wave-like properties and sometimes particle-like properties, so which is it?

• Are quantum systems both waves and particles at the same time?
• Are they sometimes waves and sometimes particles, but never both at the same time?
• Are they neither waves nor particles, but something completely different?
• Is the whole question meaningless?

These are good questions and we do not have a completely satisfactory answer to them. All of the answers have advantages and disadvantages and we cannot conclusively rule out any of them at the moment. You should not panic about this, as being in a state of uncertainty is normal in the progress of science. For example, we currently do not have a completely satisfactory answer to how life on earth started or what dark matter is made out of. This does not make it impossible to do biology or cosmology, nor does it mean we should stop looking for answers to these questions. The situation with quantum mechanics is the same. Despite the fact that it does not give us an unambiguous answer to what the world is made of does not prevent us from using the theory to make accurate predictions, nor does it mean that we should stop asking questions about the foundations of the theory. The focus of this course is on how to use the theory. I will point out foundational questions where they occur, but I will not dwell on them. If you are interested in the foundational aspects of the theory then you can take the course “Philosophy and Foundations of Quantum Mechanics” when it is offered, watch my masters level lectures on the subject, or do an undergraduate research project with me or another faculty member on these issues.

To illustrate some of the difficulties in interpreting quantum mechanics, we will consider the famous double-slit experiment from the point of view of both classical and quantum mechanics.

Richard Feynman described this experiment as:

a phenomenon which is impossible, absolutely impossible to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. The Feynman Lectures on Physics, vol. 3, section 1.1

I have to say that I strongly disagree with this characterization. In reality, there are a few different reasonable explanations of what is going on in the double slit experiment, and we just do not know which, if any, is the correct one. There are other experiments — those to do with Bell's theorem and quantum contextuality — that are much more difficult to explain in a reasonable way. We may touch on them in module 7 if we get that far, or you can find out about them in the quantum foundations experiments mentioned above.

Nevertheless, the double slit experiment does show why it is difficult to maintain an interpretation in which quantum systems either behave naively like classical particles or classical waves, independently of how we observe them, so it is worth knowing about.