Have you heard that quantum mechanics is impossible to understand? You know what, that’s what I was told, too, when I was a student. But twenty years later, I think the reason so many people believe one cannot understand quantum mechanics is because they are constantly being told they can’t understand it. But if you spend some time with quantum mechanics, it’s not remotely as strange and weird as they say. The strangeness only comes in when you try to interpret what it all means. And there’s no better way to illustrate this than the tunnel effect, which is what we will talk about today.
Before we can talk about tunneling, I want to quickly remind you of some general properties of wave-functions, because otherwise nothing I say will make sense. The key feature of quantum mechanics is that we cannot predict the outcome of a measurement. We can only predict the probability of getting a particular outcome. For this, we describe the system we are observing – for example a particle – by a wave-function, usually denoted by the Greek letter Psi. The wave-function takes on complex values, and probabilities can be calculated from it by taking the absolute square.
But how to calculate probabilities is only part of what it takes to do quantum mechanics. We also need to know how the wave-function changes in time. And we calculate this with the Schrödinger equation. To use the Schrödinger equation, you need to know what kind of particle you want to describe, and what the particle interacts with. This information goes into this thing labeled H here, which physicists call the “Hamiltonian”.
To give you an idea for how this works, let us look at the simplest possible case, that’s a massive particle, without spin, that moves in one dimension, without any interaction. In this case, the Hamiltonian merely has a kinetic part which is just the second derivative in the direction the particle travels, divided by twice the mass of the particle. I have called the direction x and the mass m. If you had a particle without quantum behavior – a “classical” particle, as physicists say – that didn’t interact with anything, it would simply move at constant velocity. What happens for a quantum particle? Suppose that initially you know the position of the particle fairly well, so the probability distribution is peaked. I have plotted here an example. Now if you solve the Schrödinger equation for this initial distribution, what happens is the following.
The peak of the probability distribution is moving at constant velocity, that’s the same as for the classical particle. But the width of the distribution is increasing. It’s smearing out. Why is that?
That’s the uncertainty principle. You initially knew the position of the particle quite well. But because of the uncertainty principle, this means you did not know its momentum very well. So there are parts of this wave-function that have a somewhat larger momentum than the average, and therefore a larger velocity, and they run ahead. And then there are some which have a somewhat lower momentum, and a smaller velocity, and they lag behind. So the distribution runs apart. This behavior is called “dispersion”.
Now, the tunnel effect describes what happens if a quantum particle hits an obstacle. Again, let us first look at what happens with a non-quantum particle. Suppose you shoot a ball in the direction of a wall, at a fixed angle. If the kinetic energy, or the initial velocity, is large enough, it will make it to the other side. But if the kinetic energy is too small, the ball will bounce off and come back. And there is a threshold energy that separates the two possibilities.
What happens if you do the same with a quantum particle? This problem is commonly described by using a “potential wall.” I have to warn you that a potential wall is in general not actually a wall, in the sense that it is not made of bricks or something. It is instead just generally a barrier for which a classical particle would have to have an energy above a certain threshold.
So it’s kind of like in the example I just showed with the classical particle crossing over an actual wall, but that’s really just an analogy that I have used for the purpose of visualization.
Mathematically, a potential wall is just a step function that’s zero everywhere except in a finite interval. You then add this potential wall as a function to the Hamiltonian of the Schrödinger equation. Now that we have the equation in place, let us look at what the quantum particle does when it hits the wall. For this, I have numerically integrated the Schrödinger equation I just showed you.
The following animations are slow-motion compared to the earlier one, which is why you cannot see that the wave-function smears out. It still does, it’s just so little that you have to look very closely to see it. It did this because it makes it easier to see what else is happening. Again, what I have plotted here is the probability distribution for the position of the particle.
We will first look at the case when the energy of the quantum particle is much higher than the potential wall. As you can see, not much happens. The quantum particle goes through the barrier. It just gets a few ripples.
Next we look at the case where the energy barrier of the potential wall is much, much higher than the energy of the particle. As you can see, it bounces off and comes back. This is very similar to the classical case.
The most interesting case is when the energy of the particle is smaller than the potential wall but the potential wall is not extremely much higher. In this case, a classical particle would just bounce back. In the quantum case, what happens is this. As you can see, part of the wave-function makes it through to the other side, even though it’s energetically forbidden. And there is a remaining part that bounces back. Let me show you this again.
Now remember that the wave-function tells you what the probability is for something to happen. So what this means is that if you shoot a particle at a wall, then quantum effects allow the particle to sometimes make it to the other side, when this should actually be impossible. The particle “tunnels” through the wall. That’s the tunnel effect.
I hope that these little animations have convinced you that if you actually do the calculation, then tunneling is half as weird as they say it is. It just means that a quantum particle can do some things that a classical particle can’t do. But, wait, I forgot to tell you something...
Here you see the solutions to the Schrödinger equation with and without the potential wall, but for otherwise identical particles with identical energy and momentum. Let us stop this here. If you compare the position of the two peaks, the one that tunneled and the one that never saw a wall, then the peak of the tunneled part of the wave-function has traveled a larger distance in the same time.
If the particle was travelling at or very close by the speed of light, then the peak of the tunneled part of the wave-function seems to have moved faster than the speed of light. Oops.
What is happening? Well, this is where the probabilistic interpretation of quantum mechanics comes to haunt you. If you look at where the faster-than light particles came from in the initial wave-function, then you find that they were the ones which had a head-start at the beginning. Because, remember, the particles did not all start from exactly the same place. They had an uncertainty in the distribution.
Then again, if the wave-function really describes single particles, as most physicists today believe it does, then this explanation makes no sense. Because then only looking at parts of the wave-function is just not an allowed way to define the particle’s time of travel. So then, how do you define the time it takes a particle to travel through a wall? And can the particle really travel faster than the speed of light? That’s a question which physicists still argue about today.
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