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The revolutionary new idea of quantum mechanics is that reality evolves through a game of chance described by probabilities. In today’s episode of Your Daily Equation, Brian explains the origin of this iconic quality and describes the equation that expresses it mathematically.
Brian Greene is a professor of physics and mathematics at Columbia University, and is recognized for a number of groundbreaking discoveries in his field of superstring theory. His books, The Elegant Universe, The Fabric of the Cosmos, and The Hidden Reality, have collectively spent 65 weeks on The New York Times bestseller list.Read More
Hey, everyone. Welcome to Your Daily Equation. Perhaps the single greatest revolution in our understanding of the universe was the move that happened in the early part of the 20th century, taking us from the older classical view of physics of the universe to the new quantum perspective. And within that change, the most radical shift in our perspective came from the need to think about the world in terms of probabilities. And indeed that is the focus of today’s Daily Equation. I’m going to motivate and then write down the basic equation that articulates mathematically this idea that the best we can ever do is predict the likelihood or the probability of getting one or another outcome, not the Newtonian perspective in which you tell me how things are today, and I will predict how they will be tomorrow with 100% certainty if I have 100% accurate data and 100% accurate calculations. Quantum mechanics says, that’s just the wrong way of thinking about things. The best you can do is predict the probability of getting one or another outcome.
All right. Now to see where this all comes from, let me just put our minds back into the right frame of mind, the right perspective. And easiest way to do that is to bring back up if you will, the double slit experiment, the famous double slit experiment in which, in this case you’re firing little particles, little electrons at a barrier with two openings, two slits. And you see that over time, the individual electrons are not landing in two bands aligned with the two openings, which is what you would have thought would happen. But instead you’re getting many bands that are spread out along the detector screen, getting this iconic pattern of bright regions, dark regions, bright regions and dark regions.
And it was realized after some consternation that this data is strongly hinting at the fact that when you describe the quantum mechanical motion of particles, you need to think about them in terms of waves. Why is that? Well, look, here’s a nice little image that describes wave motion. And you see that you have waterways heading toward a barrier with the two openings. Out of each opening, you have circular water waves radiating, and because they are intermingling, they’re overlapping, you get this interesting pattern, places where the peak of one wave crosses the trough of another, cancel each other out. And that’s why you’ve got regions of high agitation, no agitation, high agitation and so forth as you look across. And on that back screen, you can see here, you’ve got, if I let brightness on that screen register the agitation of the water, then you’ve got bright, dark, bright, dark, bright, dark just as we had with the data of particles being fired at the screen with the two openings.
And this is where the link between particles and waves becomes manifest. Same kind of data, even though it appears to be that you have different things being thrown at the barrier. Particles in one case, waves in another, same data suggested deep connection between them. And indeed there is a deep connection.
But people struggled in the 1920s to figure out exactly what that connection would be. I mean, look, you know a particle, it’s something that exists at a little tiny location in space. That’s what a particle is. It’s like the smallest speck of something. A wave on the other hand is completely different. It’s completely spread out. Kind of the opposite of being localized at one special point. And so how do you link these two ideas that seem to be at opposite ends of that particular spectrum? And many people weighed in in the 1920s. Erwin Schrodinger, I’ll show you a little picture of him for a little cultural context, a little visualization here. Schrodinger, who’s one of the dominant figures in quantum mechanics, there’s a famous equation that many of you have asked me to talk about, and I will talk about it, not today, but very soon I will, Schrodinger’s equation. But Schrodinger weighs in with the possible way in which a particle might be a wave, and he says, look, he says maybe a particle like an electron, in reality you think of it as a little dot, but maybe it’s actually possible that you can have a kind of smeared out electron kind of like peanut butter or mayonnaise. You can smear it out. Maybe an electron can be smeared out too. And that’s the way an electron can be a wave. It’s not really only at a point. It can be spread out.
Now it’s a natural suggestion, but as people thought about it, it ran up against a formidable hurdle, which is, look, if an electron really can be spread out in space, then in principle you should be able to measure an electron and and get a fraction of an electron. You should be able to grab a little piece of that smeared out electron, have a 10th of an electron or a quarter of an electron or a half of an electron. But the data consistently showed and shows that when you measure an electron, you always get an entire electron, the whole thing. There doesn’t seem to be any sense to be made of a 10th of an electron or a third of an electron.
So that interpretation of the wave associated to a particle as just a kind of smeared out mushy spread out particle doesn’t work. So people continue to put forward ideas. But it was actually this guy right here. This is Max Born, who is a somewhat lesser known figure, at least to the general public when you think about the pantheon of physicists that developed quantum mechanic, kind of a mild mannered sort of guy who would quietly work in the shadows of some of the other giants. And in this particular case, the idea that he put forward is at the core of quantum mechanics. It is the profound interpretive idea that comes through Max Born where he says, look, what’s the wave associated with a particle like an electron? He says it’s a kind of wave that you have never encountered before. It involves an idea that seems completely distinct from the typical ideas that we invoke when we talk about the motion of particles. He says the wave is a wave of probability. A wave of probability, a probability wave.
What does that mean? Well, it means that when you talk about the motion of an electron, there is a wave, and where the wave is big, roughly speaking, I’ll be more precise in a moment. Where the wave is big, there’s a high probability that the electron will be found. Where the wave is small, there’s a smaller probability that you will find the electron at that location. And where the wave is zero, like where the crisscrossing waves out of the double slit experiment where the way from one side can cross away from the other and cancel each other out, where the probability wave is zero, there is a zero chance that you will find the electron. And from this interpretive scheme, the dark regions on the interference pattern on that detector screen, those would be places where the probability waves have canceled each other out from both openings. The probability of the particle being found there is therefore zero. And that’s why that region on the detector screen is dark. The bright regions would be where the probability of finding the electron is high. So when you throw many electrons over time, many will land in those locations were it’s highly probable that they will land. And that’s his way of explaining the wavelike nature of matter and also how, for instance, you would use it to explain the data in the double slit experiment.
Let me show you a little visual on that. And so I’m going to use a kind of blue little wave as you see here to represent the wave associated to a particle, like an electron probability wave. And again, where the height of the wave is big, intuitively you can think about that as a location where it’s likely to find the particle. Where it’s smaller, less likely. And places where the wave is zero, you won’t find the particle.
Now imagine you undertake a measurement of the position of the electron. Boom. And you find it at one or another location. What does that mean? Well, it means that you now know where the electron is with certainty because you’ve measured its position, which means the wave must spike as we see here where all of the probability is now centered on that location. And the wave is zero elsewhere because by virtue of your measurement, you know it is not located at any of those other locations, and therefore the probability of finding it at those locations must drop all the way to zero.
Now I should say that this process of measuring a particle and the quantum wave responding as I’ve shown in this little visual, snapping to attention and spiking at the location where you’ve found the particle, this is a highly controversial part of quantum mechanics. We will come back to it. This is called the collapse of the wave function. Wave function is the more precise name for the probability wave associated to a particle. We are far from certain that this kind of spiking, this kind of collapse actually happens. In fact, I would say that most people think that it doesn’t happen in this naive way that I’ve described it, but for pedagogical reasons, it’s useful to think about things at first in this language. But how you go from the nebulous probabilities to the definite reality that emerges from an observation is one of the big unsolved problems in quantum mechanics, the so called quantum measurement problem.
But putting that problem to the side, the idea, if we go back to say the way people were thinking about things in the 1920s, is after you performed the measurement, and the wave has this spike shape, if you then look away, that spike shape will kind of melt back into this wavy probabilistic landscape of possibilities until the next measurement happens, which causes this fuzzy haze of possibilities to again snap to attention and give rise to the single definite reality that your observation or your measurement reveals.
Now look, I should stress this is a bizarre, highly counter-intuitive, very strange way of talking about the world. We are unused to a reality that when we’re not measuring, when we’re not looking, is a fuzzy haze of many possibilities whose probabilities are described by some quantum mechanical undulating wave. And only when we look at the world does it then snap to attention and have the definite qualities that we all experience every moment of our lives. But this is the new image that is emerging from this wavelike nature of matter revealed today by the double slit experiment in the hands of Max Born with this probabilistic interpretation of what the wave actually means.
Now let me just put a little bit of math equation behind this all. After all this is called Your Daily Equation. It is not called your daily animation or your daily imagery. So let me just show you. Let me bring this up on the screen. There we go. Good. Okay, so first off you could easily ask the question straight off the bat. You could say, look, when we have a wave, sort of a wave that looks like this, for instance, there are places where the wave is a positive and places where it’s negative. So it can’t literally be the wave itself that is giving you the value of the probability because if that were the case, you’d be faced with a situation, say of having negative probabilities. What in the world would that mean? And indeed, in the interpretive scheme that Born gives us, it is not literally the size of the wave at a given location that tells you the probability. It’s a little bit more complicated. It’s further complicated by a word that’s actually related to complicated, complex, and this is a mathematical detail that no doubt some of you are familiar with. But let me just quickly say it.
The actual wave in quantum mechanics, it’s often described using the Greek letter psi. And we say, psi of X and T. If we’re doing one dimensional non-relativistic quantum mechanics, so I only have one coordinate over here. Imagine I’m just doing quantum mechanics on a line as opposed to space just for ease. This is actually, in general, as we will see when we talk about Schrodinger’s equation, this is actually a complex number. Remember complex numbers involving square root of minus one which is equal to i. So if it’s actually a complex number, then the issue of having negative probabilities is even accentuated now because now you’ve got imaginary probability. So what do you do with that? So the answer is, the final answer that Born gives is the probability, say a finding the electron at position X, you don’t set it equal to the wave itself. And let’s say this is at a given moment in time, maybe I should put that in here to avoid any confusion. You don’t set it precisely directly equal to the value of the wave. Rather you take that wave, and you multiply it by its complex conjugate, which we often write us psi of XT squared. And the beauty of this particular combination of the wave times its complex conjugate is that that product is always a real number, and it’s always a positive number, so it’s good. So our probabilities are real and positive just like we’d want.
Now the other thing, just as a mathematical nicety, you want it to be the case that if you add up all of the probabilities of where you will find a particle, say at a given moment in time, let’s just hold the time fixed. And I’ll indicate that by having little zero on it. The sum of all the probabilities should add up to one. That’s the definition really of what probabilities are. So the little mathematical detail that’s just worth writing down here is when we talk about these wave functions, we demand that we normalize them in such a way that if, for instance, I integrate, which is the continuous version of taking a sum, this guy here at any given moment in timer, if I integrate over the domain where the particle might be, it could be from minus infinity to infinity. It could be over a finite interval that the sum of those probabilities is equal to one as we have here.
And I can enforce that by just renormalizing the value of the wave function by scaling it by a constant value, namely the square root of whatever number this integral was, if it was not equal to one. So if I properly normalize my probability wave, my wave function, then we come to this interpretation that the probability of finding the particle is given by the norm squared of the probability wave at that location at that moment in time.
And that is the interpretive scheme that Max Borne gives us back in the 1920s. It is a profound move. Again, as we started out going from the Newtonian picture where what you see is what there is. You describe the stuff that’s out there. And it’s a definite reality, and your math describes a definite reality. But Max Born, together with all of the others who were developing this quantum picture says, no, what’s actually out there, quantum mechanical waves, those waves evolve in a manner that, as we will discuss maybe next time, maybe the time after, is given by an equation, Schrodinger’s equation. So we’ll come to that, but these wave evolve through time. And what they tell you is the probability of finding this or that particle at this or that location. And that’s the best you can do. It’s not that you’re missing something. And for that reason, you’re relegated to talk in terms of probabilities. It’s not like the weatherman or flipping a coin in the classical sense that we just don’t have enough information to tell you exactly what’s going to happen. So our ignorance forces us into the language of probabilities.
This, at least in one interpretation, there are other interpretations of these ideas, maybe I’ll talk about some of them, but in the most straightforward interpretation, that’s not what’s going on. What’s going on is the world is fundamentally probabilistic. You’re not missing anything. You’re not ignorant. The best you can do is describe the likelihood of one or another outcome. And it is this quantum mechanical probability wave that captures those probabilities mathematically.
Okay. That’s what I wanted to talk about today. This key move in quantum mechanics to the probabilistic equation, giving us a mathematical version of describing the world in terms of probabilities, and we’ll carry on with our discussion of quantum mechanics in subsequent episodes. But for today, that’s it. This is Your Daily Equation. Take care until next time.