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PROFESSOR: So this brings us to the conclusion of our Massive Open Online Course, 10.S9,5 Physics of COVID-19 Transmission. So let's briefly review what we've learned. We began by talking about respiratory pathogens, both bacteria and viruses. And, in particular, we focused on viruses and understood how they can be transmitted through aerosol particles through infected indoor air. And this includes the important case of SARS-CoV-2, the virus that causes COVID-19. We then went on to use epidemiological models and fluid-mechanical analysis of a well-mixed room to arrive at a universal indoor safety guideline to limit transmission of the disease. A very important conclusion is that it's not possible to bound a single variable, as in all of the current safety guidelines from various official organizations. So, for example, you cannot only bound the distance between people, for example, 6 feet, 1 meter. You cannot bound only the occupancy, say, 25 persons. You can...

PROFESSOR: That brings us to claim number four, which is perhaps the most important one. I may have said it already.b The eigenfunctions of Q form a set of basis functions, and then any reasonable psi can be written as a superposition of Q eigenfunctions. OK, so let's just make sense of this. Because not only, I think we understand what this means, but let's write it out mathematically. So the statement is any psi of x, or this physical state, can be written as a superposition of all these eigenfunctions So there are numbers, alpha 1 psi 1 of x plus alpha 2 psi 2 of x. Those are the expansion coefficients with alphas. And in summary, we say from sum over i, alpha i psi i of x. So the idea is that those alpha i's exist and you can write them. So any wave function that you have, you can write it in a superposition of those eigenfunctions of the Hermitian operator. And there are two things to say here. One is that, how would you calculate those alpha i's? Wel...