The most common question you ask yourself when studying quantum physics is this: what are those amplitudes, and why do we have to square them to get probabilities?
It is a question which cannot easily be answered because it depends on what we are modeling: a two-state system, electron orbitals, something else? And if it is a two-state system, is it a laser, some oscillation between two polarization states, or what? These are all very different physical systems and what an amplitude actually is in that context will, therefore, also be quite different. Hence, the professors usually just avoid the question and just brutally plow through all of it all. And then, by the time they are done, we are so familiar with it that we sort of forget about the question. [I could actually say the same about the concept of spin: no one bothers to really define it because it means different things in different situations.]
In fact, I myself sort of forgot about the question recently, and I had to remind myself of the (short) answer: probabilities are proportional to energy or mass densities (think of a charge spending more time here than there, or vice versa), and the energy of a wave or an oscillation – any oscillation, really – is proportional to the square of its amplitude.
Is that it? Yes. We just have to add an extra remark here: we will take the square of the absolute value of the amplitude, but that has got to do with the fact that we are talking oscillations in two dimensions here: think of an electromagnetic oscillation—combining an oscillating electric and magnetic field vector.
Let us throw in the equations here: the illustration below shows the structural similarity (in terms of propagation mechanism) between (1) how an electromagnetic wave propagates in space (Maxwell’s equations without charges) and (2) how amplitude waves propagate (Schroedinger’s equation without the term for the potential resulting from the positive nucleus). It is the same mechanism. That is actually what led me to the bold hypothesis, in one of my very first papers on the topic, that they must be describing the same thing—and that both must model the energy conservation principle—and the conservation of linear and angular (field) momentum!
Is that it? Is that all there is? Yep! We have written a lot of papers, but all they do is further detail this principle: probability amplitudes, or quantum-mechanical probability amplitudes in general, model an energy propagation mechanism.
Of course, the next question is: why can we just add these amplitudes, and then square them? Is there no interference term? We explored this question in our most recent paper, and the short answer is: no. There is no interference term. [This will probably sound weird or even counter-intuitive – it is not what you were thought, is it? – but we qualify this remark in the post scriptum to this post.]
Frankly, we would reverse the question: why can we calculate amplitudes by taking the square root of the probabilities? Why does it all work out? Why is it that the amplitude math mirrors the probability math? Why can we relate them through these squares or square roots when going from one representation to another? The answer to this question is buried in the math too, but is based on simple arithmetic. Note, for example, that, when insisting base states or state vectors should be orthogonal, we actually demand that their squared sum is equal to the sum of their squares:
(a + b)2 = a2 + b2 ⇔ = a2 + b2 = a2 + b2 + 2a·b ⇔ a·b = 0
This is a logical or arithmetic condition which represents a physical condition: two physical states must be discrete states. They do not overlap: it is either this or that. We can then add or multiply these physical states – mix them so as to produce logical states, which express the uncertainty in our mind (not in Nature!), but we can only do that because these base states are, effectively, independent. That is why we can also use them to construct another set of (logical) base vectors, which will be (linearly) independent too! This will all sound like Chinese to you, of course. Any case, the basic message is this: behind all of the hocus-pocus, there is physics behind—and so that’s why I writing all of this. I write for people like me and you: people who want to truly understand what it is all about.
My son had to study quantum physics to get a degree in engineering—but he hated it: reproducing amplitude math without truly understanding what it means is no fun. He had no time for any of my detailed analysis and interpretation, but I think I was able to help him by telling him it all meant something real. It helped him to get through it all, and he got great marks in the end! And, yes, of course, I told him not to bother the professor with all of my weird theories! 🙂
I hope this message encourages you in very much the same way. I sometimes think I should try to write some kind of Survival Guide to Quantum Physics, but then my other – more technical – blog is a bit of that, so that should do! 🙂
PS: As for there being no interference term, when adding the amplitudes (without the 2a·b term), we do have interference between the a and b terms, of course: we use boldface letters here but these ‘vectors’ are quantum-mechanical amplitudes, so they are complex-valued exponentials. However, we wanted to keep the symbolism extremely simple in this post, and so that’s what we did. Those wave equations look formidable enough already, don’t they? 🙂