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I have just told you that quantum theory is a different kind of probability theory. So the obvious question to ask, how different it is, why do we need quantum theory in order to calculate probabilities? Well the short answer of course is that the nature doesn't quite follow the classical probability theory, but let's see where the difference really is. So consider the physical system that evolves from some configuration A to some configuration B. Let me just call those configuration states. So the system that evolves from state A to state B. And let's assume that there are two alternatives for the system to evolve from A to B, and let's associate probabilities P1 and P2 with those alternatives.

Then at this stage, you would say, well, if I ask you a question, what's the probability that this process will evolve? will happen, so the system will evolve from A to B, then you will say, well, of course, we know how to calculate probabilities using the classical theory of probability, and in which case you just simply follow or refer to the Kolmogorov Additivity Axiom, which says that if something can happen in two mutually exclusive ways, you simply add up probabilities associated with each way separately. So you're probably good. probability P, according to the classical theory of probability, is simply P1 plus P2. So that's what classical theory of probability is telling you.

How about quantum physics? So quantum theory tells you that, well, first of all, we have to actually find probability amplitudes associated with those two different alternatives. The quantum theory per se doesn't give you those numbers, but you may just rely on some other knowledge of quantum mechanics. physical interactions involved and find numbers, find probability amplitudes, let me call them alpha 1 and alpha 2, that correspond to those two different alternatives. So the system evolves from A to B and can follow this particular path with probability amplitude alpha 1 or this particular path with probability alpha 2. And we know that the quantum theory tells us that we have to start by adding probability amplitudes, not probability. probabilities.

So probability amplitude for the whole process alpha is equal to alpha 1 plus alpha 2. Now, the quantum physics also tells us that if we want to calculate probability we have to take the mod square of alpha. And when we take the mod square of alpha, that is equal to the mod square of alpha 1 plus alpha 2. And then when we expand this expression here, we will get alpha 1 mod square plus alpha 2 mod square. And then, and here it becomes, it's getting interesting here, we're going to get the cross terms, namely alpha 1, alpha 2 complex conjugate plus alpha 1 complex conjugate alpha 2. And in order to see this. what this term really looks like.

We'll focus on this term for a moment. Let me just express alpha 1 and alpha 2 in the polar form. So alpha 1 will be equal to mod alpha 1 EI phi 1, some phase factor, and alpha 2 will be equal to mod alpha 2 EI phi 2. So when I substitute or express my probability amplitudes in the polar form, I get this. So I can write this as mod alpha 1 EI phi 1, and I can write this as mod alpha 2 EI phi 2. So I can write this as mod alpha 1 EI phi 1, and I can write this as mod alpha 2 EI phi 2.

So I can write this as mod alpha 1 EI phi 1, and I can write this as mod alpha 2 EI phi 2. So I can write this as mod alpha 1 EI phi 1, and I can write this as mod alpha 2 EI phi 2. So I can write this as mod alpha 1 EI phi 1, and I can write this as mod alpha 2 EI phi 2. So I can write this as mod alpha 1 EI phi 1, and I can write this as mod alpha 2 EI phi 2.

So I can write this as mod alpha 1 EI phi 1, and I can polar form, then I will get alpha 1 mod squared plus alpha 2 mod squared plus 2 alpha 1, the absolute value of alpha 1, absolute value of alpha 2, and I'll have a cos phi 2 minus phi 1, which is a very simple calculations. So let me then just perhaps write it as p1, so if you look at this expression alpha one mod square, that's the probability that the system follows the first path, right? Plus p2, clearly you can see that this term here corresponds to the probability that the system follows this path. And here comes the interesting part.

We can write this the remaining term here as 2 times p1 times p2 cos phi2 minus phi1. Now let's look at this expression here. We can recognize the first two terms p1 plus p2 as a something that we are familiar with, right? Because that's really the classical part. That's what classical theory of probability tells you that we have to find. add probabilities. But there's more. So you see the quantum theory adds one extra expression to the whole thing. And this extra expression here, which is called quantum interference terms, can be both positive or negative because the cost can go from minus one to plus one.

So therefore the interference term, the quantum interference terms, can enhance probability for the system to go from A to B. Or, depending on the phase difference here, can actually diminish this probability. So you can see that somehow the phases, the phase factors that are involved, control the degree to which we can enhance or reduce the probability as compared to the classical part. The other thing that is very interesting here is the fact that we have the difference between the phases, phi2 and phi1. So some. Somehow the system, evolving from A to B, knows about the phase differences. That's something, you know, it's kind of trivial from the mathematical point of view.

It just pops up this difference, right? But for physicists, it's another trivial thing, because now when you ask yourself the question, how come the system knows about the phase setting on one part and the other one so that it can actually react to the difference in the phase setting? So the physicists at this point, depending on whom you are talking to, can give you all kinds of different answers. But most would agree that somehow the system goes into a state that it follows in some sense at least the two paths at the same time. And those who then want to sort of defend Kolmogorov and say, ah, then perhaps it is not the case that those are two mutually exclusive.

mutually exclusive events, that those are perhaps not really alternatives. Well, indeed, so that may be one way of thinking about it. That is certainly not the case that the system goes either this way or this way. Those are not mutually exclusive things. In the quantum way, the system can actually propagate from A to B, feeling the two paths at the same time. It's clearly as it is. than here because it reacts to the difference in the phase setting.

So how could it possibly react to the difference without being both here and there? We're going to explore this fact a lot in our lectures, but for now do remember that the way quantum physics sort of differs from the classical probability theory is by this one extra interference term that could be positive or negative, and therefore we can increase or enhance probabilities and reduce probabilities as compared to the classical case. .