Amplitudes and probabilities

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!

Energy waves

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·ba·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·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? 🙂

Uncertainty, quantum math, and A(Y)MS

This morning, one of my readers wrote me to say I should refrain from criticizing mainstream theory or – if I do – in friendlier or more constructive terms. He is right, of course: my blog on Feynman’s Lectures proves I suffer from Angry Young Man Syndrome (AYMS), which does not befit a 50-year old. It is also true I will probably not be able to convince those whom I have not convinced yet.

What to do? I should probably find easier metaphors and bridge apparent contradictions—and write friendlier posts and articles, of course! 🙂

In my last paper, for example, I make a rather harsh distinction between discrete physical states and continuous logical states in mainstream theory. We may illustrate this using Schrödinger’s thought experiment with the cat: we know the cat is either dead or alive—depending on whether or not the poison was released. However, as long as we do not check, we may describe it by some logical state that mixes the ideas of a dead and a live cat. This logical state is defined in probabilistic terms: as time goes by, the likelihood of the cat being dead increases. The actual physical state does not have such ambiguity: the cat is either dead or alive.

The point that I want to make here is that the uncertainty is not physical. It is in our mind only: we have no knowledge of the physical state because we cannot (or do not want to) measure it, or because measurement is not possible because it would interfere (or possibly even destroy) the system: we are usually probing the smallest of stuff with the smallest of stuff in these experiments—which is why Heisenberg himself originally referred to uncertainty as Ungenauigkeit instead of Unbestimmtheit.

So, yes, as long as we do not look inside of the box – by opening or, preferably, through some window on the side (the cat could scratch you or jump out when opening it) – we may think of Schrödinger’s cat-in-the-box experiment as a simple quantum-mechanical two-state system. However, it is a rather special one: the poison is likely to be released after some time only (it depend on a probabilistic process itself) and we should, therefore, model this time as a random variable which will be distributed – usually more or less normally – around some mean. The (cumulative) probability distribution function for the cat being dead will, therefore, resemble something like the curves below, whose shapes depend not only on the mean but also on the standard deviation from the mean.

1920px-Normal_Distribution_CDF

Schrödinger’s cat-in-the-box experiment involves a transition from an alive to a dead state: it is sure and irreversible. Most real-life quantum-mechanical two-state systems will look very different: they will not involve some dead-or-alive situation but two very different states—position states, or energy states, for example—and the probability of the system being in this or that physical state will, therefore, slosh back and forth between the two, as illustrated below.

Probabilities desmos

I took this illustration from the mentioned paper, which deals with amplitude math, so I should refer you there for an explanation of the rather particular cycle time (π) and measurement units (ħ/A). The important thing here – in the context of this blog post, that is – is not the nitty-gritty but the basic idea of a quantum-mechanical two-state system. That basic idea is needed because the point that I want to make here is this: thinking that some system can be in two (discrete) physical states only may often be a bit of an idealization too. The system or whatever is that we are trying to describe might be in-between two states while oscillating between the two states, for example—or we may, perhaps, not be able to define the position of whatever it is that we are tracking—say, an atom or a nucleus in a molecule—because the idea of an atom or a nucleus might itself be quite fuzzy.

To explain what fuzziness might be in the context of physics, I often use the metaphor below: the propeller of the little plane is always somewhere, obviously—but the question is: where exactly? When the frequency of going from one place to another becomes quite high, the concept of an exact position becomes quite fuzzy. The metaphor of a rapidly rotating propeller may also illustrate the fuzziness of the concept of mass or even energy: if we think of the propeller being pretty much everywhere, then it is also more useful to think in terms of some dynamically defined mass or energy density concept in the space it is, somehow, filling.

propeller This, then, should take some of the perceived harshness of my analyses away: I should not say the mainstream interpretation of quantum physics is all wrong and that states are either physical or logical: our models may inevitably have to mix a bit of the two! So, yes, I should be much more polite and say the mainstream interpretation prefers to leave things vague or unsaid, and that physicists should, therefore, be more precise and avoid hyping up stuff that can easily be explained in terms of common-sense physical interpretations.

Having said that, I think that only sounds slightly less polite, and I also continue to think some Nobel Prize awards did exactly that: they rewarded the invention of hyped-up concepts rather than true science, and so now we are stuck with these things. To be precise, I think the award of the 1933 Nobel Prize to Werner Heisenberg is a very significant example of this, and it was followed by others. I am not shy or ashamed when writing this because I know I am in rather good company thinking that. Unfortunately, not enough people dare to say what they really think, and that is that the Emperor may have no clothes.

That is sad, because there are effectively a lot of enthusiastic and rather smart people who try to understand physics but become disillusioned when they enroll in online or real physics courses: when asking too many questions, they are effectively told to just shut up and calculate. I also think John Baez’ Crackpot Index is, all too often, abused to defend mainstream mediocrity and Ivory Tower theorizing. At the same time, I promise my friendly critic I will think some more about my Angry 50-Year-Old Syndrome.

Perhaps I should take a break from quantum mechanics and study, say, chaos theory, or fluid dynamics—something else, some new math. I should probably also train to go up Mont Blanc again this year: I gained a fair amount of physical weight while doing all this mental exercise over the past few years, and I do intend to climb again—50-year-old or not. Let’s just call it AMS. 🙂 And, yes, I should also focus on my day job, of course! 🙂

However, I probably won’t get rid of the quantum physics virus any time soon. In fact, I just started exploring the QCD sector, and I am documenting this new journey in a new blog: Reading Einstein. Go have a look. 🙂

Post scriptum: The probability distribution for the cat’s death sentence is, technically, speaking a Poisson distribution (the name is easy to remember because it does not differ too much from the poison that is used). However, because we are modeling probabilities here, its parameters k and λ should be thought of as being very large. It, therefore, approaches a normal distribution. Quantum-mechanical amplitude math implicitly assumes we can use normal distributions to model state transitions (see my paper on Feynman’s Time Machine).