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  • Writer's pictureNoa Feldman

2 - The uncertainty principle

Is it your first visit here? Welcome! My name is Noa and I am a Physics Ph.D. student at Tel Aviv University. I write here about quantum mechanics for non-physicists. No background in mathematics or physics is required to read the blog, but I highly recommend reading the first post before reading this one.

As we have seen in the last post, the quantum world is a very strange one. A physical body does not have a definite position, but rather a distribution of positions, which we call a superposition. Only if we observe a body, do we make it ‘choose’ - or collapse - into a definite position. This means that we make things happen only by observing them. It feels as if this world is completely different from the classical world in which we live our day-to-day lives. By the end of this post, you will see that the quantum and classical worlds are indeed the same world.

We start by noticing something important: We can measure (observe) the position of all of the bodies in the world. After the measurement, the position of the body collapses. After collapsing, it has a definite position, as we know and love from the classical world. So if we just go to all of the bodies in the universe and measure their position, everything will collapse and stop being quantum, right?

If only things were that simple.

What is Uncertainty?

First of all, let us discuss a small inaccuracy in the description of superposition we saw in the last post. We looked at a ball and measured its position using a ruler. However, we cannot measure the position exactly using a ruler - we can only decrease the uncertainty in the position of the ball. If the ruler is 30cm and the screen covers its entire length, we can say that the uncertainty in the position of the ball is as long as the screen - 30cm.

After we moved the screen and measured the ball’s position, we may think that we know exactly where the ball is - say 22.5cm relative to the ruler.

If we look very closely, we may be able to say that in fact, the center of the ball is closer to 22.4cm relative to the ruler, but this will not be accurate as well. If we wanted to know the position of the ball up to an accuracy of 0.1mm, we would not be able to tell. If we want to know the position up to an accuracy of nanometers - 1 billionth of a meter - we would have no idea what to answer at all, so much so that it would be like having an entire screen covering our ball in the nanometer-scaled ruler. In short, we still don’t know the position of the ball for sure, we have just narrowed the uncertainty regarding its position from 30cm to around 1mm.

This is always the case. When we look at physical bodies, we can only tell their position up to the sensitivity of our eyes. If we take a better detector than our eyes, like a high-resolution camera, we can only tell the position up to the resolution of the camera. There will always be some uncertainty.

The uncertainty principle

The superposition of the ball was manifested by some probability distribution, and by moving the screen we have changed this distribution. In the quantum world, measurement changes the state of the body. So when we performed a measurement of the position, we changed the distribution of the position of the ball and decreased the uncertainty. But the ball’s state is determined by more distributions than position: it has a velocity (speed) distribution, energy distribution, and so on. This means that we don’t only have uncertainty regarding the position of the ball, but we also have uncertainty regarding the speed that the ball can move at.

If a measurement changes the distribution of the position, what keeps it from changing the distribution of the velocity as well? It turns out that this is exactly what it does. The position of the body and the speed of the body are called dual properties. It means, in short, that when we perform a measurement and decrease the uncertainty in position, we increase the uncertainty in velocity. In other words, the more we know where our ball is, the less we know what its speed is!

(In fact, the dual property of position is momentum and not velocity. These two properties are pretty similar for most of the bodies we know, so it is just fine to think about speed instead if you don’t know what momentum is exactly).

The property we just saw is called Heisenberg’s uncertainty principle. It means that we can never know for sure both the position and the speed of a quantum particle. Mathematically, it is put this way:

Let’s see what each of these terms mean:

is the uncertainty in position. In our case, it starts at about 30cm and then changes to around 1mm.

is the uncertainty in momentum, which we treat as the uncertainty in speed.

is some constant of nature, called Planck's constant named after one of the founders of modern physics, Max Planck. For simplicity, let us say that it equals 2 cm^2/second. Its real value is more complicated, and we will get to it in a minute.

The uncertainty principle means that the multiplication of the uncertainties, position and speed, cannot exceed Planck's constant divided by 2, which is 1 cm^2/second. In our example, at first, the position uncertainty was 30cm, so the speed uncertainty had to be at least 1/30 cm/second. After the measurement, the position uncertainty became much smaller - 0.1cm. Now the speed uncertainty needs to be much larger - at least 10 cm/second. Observing the ball’s position made us know less about the speed. If we want to measure the speed, it will make the uncertainty in the position larger again. We will always have uncertainty regarding the state of our quantum ball, no matter how hard we try to avoid it.

Sewing the quantum and classical together

Based on the uncertainty principle, once we limited the possibilities of the position of the ball, we lost a lot of information regarding its velocity, so we cannot really tell where it would be in a minute anyway. How come this rule is true? In the real world, we know where things are and how fast they are moving every now and then, right? How can we not feel all of these properties in the classical world?

The key is in Planck's constant. Its real value is not 2 cm^2/second, but instead:

This means 33 zeros after the decimal point before the figures 106 show up. This is a tiny number. Note that it has units of kg - it means that the uncertainty principle depends on the mass of the bodies. Very light bodies, like electrons or atoms, are affected by the uncertainty principle in a significant way. But when we talk about heavy objects, like balls, people, or even feathers, the uncertainty is still way too small for us to feel it - this is true both for position and for speed.

The quantum properties still hold. Your position right now on the couch or on the train is not definite, but in a superposition. It’s just that this position is spread across a really really (really) small range of possible positions. Such a small range that our senses, or even the best detectors we can build, still treat this range as a single position.

If we look closely at day-to-day bodies we see that the behavior of the atoms they are composed of is quantum and peculiar. If we went further and further from them, their uncertainties would become smaller and smaller until we don’t notice them anymore and the world looks like the classical, day-to-day world that we know.

But we don’t want to stay in the classical world, where the uncertainties are so small that they don’t mean anything and nothing interesting happens. The quantum principles are relevant to light particles like atoms, electrons, or photons (particles of light) - the building blocks of our world. They help us understand electrical currents, chemical molecules, or light coming from the sun. So here in this blog, we will mostly stay in the quantum realm, where all of these things happen. Next time we will learn about an important experiment that helped confirm the quantum theory. Don't worry if this is still puzzling to you - we will see some examples and look at our new knowledge from a different angle that might help.

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