Quantum Tunneling: Explained
On the smallest scale, the world is weird. When things get down to the size of an atom, the physics that governs us no longer applies. We throw classical mechanics out the window and quantum physics takes to the stage. And since it’s quantum physics, new rules are needed, and to describe them, new counterintuitive equations. The idea of an objective reality goes out the window, replaced with notions like probability distributions rather than deterministic outcomes, wavefunctions rather than positions and momenta, Heisenberg uncertainty relations rather than individual properties.
Enter: Tunneling
Among all the successes of the quantum world, none were more impressive than the understanding of the tunnel effect. The phenomenon of tunneling, which has no counterpart in classical physics, is an important consequence of quantum mechanics. But to explore that, we’ll need to first lay some groundwork.
Classical Physics
For mathematical and physical convenience, we’ll restrict our quantum particle to a straight line. That way we only have to consider our particle’s motion in one dimension. Now, under just these limitations, technically our particle is a “free” particle. Or in other words, no external forces are acting on it. Just a particle in empty space. But let’s change that. Let’s introduce something else to the system: a potential barrier.
A potential barrier is any region of space in which particles are decelerated or stopped by a repulsive force.
We can do this by introducing another particle to our system. This second particle, however, won’t actually move. It’s fixed in space. The only reason that we even have it is to theoretically introduce a potential barrier.
So here we’ve placed two particles in a region of space. And one’s fixed while the other isn’t. We, now, blindly establish that the charges on both these particles are the same:
- positive, positive
- negative, negative
We only do this because if these two particles were opposite charges they’d attract and consequently, there’d be a force between them. A force that will favor the motion of the particle in that particular direction. But as you might recall, a potential barrier presupposes a repulsive force, not an attractive one.
Anyway, it goes unsaid that like charges repel and unlike ones attract. So the newly introduced fixed particle radiates an electric field. And this electric field interacts with that of the mobile particle’s. Since they’re the same charge, the mobile particle is decelerated.
When the mobile electron moves towards the stationary one, the magnitude of repulsive force it experiences increases. And if enough energy is not provided to this electron, it won’t be able to do work against this repulsive force that it experiences. In other words, it’ll bounce back. But if we wanted it to pass, we’d give our first electron a lot more kinetic energy. So, let’s do just that.
Even this time it’ll experience a repulsive force and as a consequence, it will slow down near the second electron. But since the kinetic energy is greater than the potential, it will pass through. We can represent this entire situation with a sort-of glorified graph. On this graph, we’re representing the line along which our first electron is traveling — x — and the energy of our electron. The graph, essentially, is a representation of the energy of our electrons at different positions in space. The first energy that we can represent is the potential energy barrier due to the repulsion of the two electrons, V(x). Now, on this graph, we can also represent the energy we are giving to the first electron, E.
Essentially, the idea is that if we don’t provide the first electron with enough energy, it’s incapable of surmounting the barrier established by the second electron. And what that means is this electron cannot get to a position to the right of the barrier. If you can, however, provide the first electron with enough energy, it can get over the potential barrier.
Now I will stop here for a moment and acknowledge that the terminology I’ve used here is kind of wishy-washy. But what I want you to take away from this is an understanding of how an electron would behave if it were restricted to a straight line. Or at least, this is what classical physics dictates. With the advent of the quantum world, however, things change.
To fully understand quantum tunneling we need to abstract things a little bit. Let’s forget our electron and the potential barrier and now introduce something which looks like this:
Now this here is essentially the same as the potential barrier we established earlier. Except for the one change that instead of a continuously changing function, V(x), we now have a function where the energy peaks suddenly. So, V is 0 before the point x = a, and the V is zero after x = b. But it is not zero between a and b.
Recalling what classical physics tells us, the electron will not pass through the barrier if it has less energy than the peak potential, u.
What we’ve done so far seems pretty simple. But if only this were that straightforward. What we haven’t taken into account this far is quantum mechanics.
What we’ve established seems fairly reasonable. Or at least if we were in the regime of the classical world. But as the title suggests, that’s not how things are done here.
What quantum mechanics tells us is that knowing the position and velocity of this electron is a lot more complicated than classical physics. And all of this is coincidentally linked to the wavefunction. The wavefunction, briefly explained, is a mathematical expression that contains all the information we can possibly know about a particle. This wave function, when squared, then tells us the probability distribution of finding our electron.
In other words, in the classical world, we knew that our electron was in one defined point in space but with quantum mechanics, things are different. Instead of establishing a particular point, all we can do is establish a probability. We can’t, ever, with certainty say that an electron is at a specific point in space. We can only say that the electron is probably there.
Ψ² ∝ probability distribution of finding a particle at a particular point.
It’s this function like we established, that is the important one. Specifically, this function is telling us — Ψ² is telling us — that at r=2 because Ψ² is large, we’re most likely to find our electron. Whereas at 4, Ψ² is small and so we’re least likely to find our electron. Now there are a few subtleties to this but for our purpose, all we care about is that this Ψ² function is telling us the probability distribution of finding our electron in our one–dimensional system. We won’t necessarily bother ourselves with the form of the wavefunction itself. All that matters is that if we wanted to, we could find the Ψ of our system.
I won’t go too deep into the probabilistic nature of quantum mechanics here. If you’d like, check these out:
Well then, there comes the question:
How do we know what the wave function of our electron looks like when E < u?
For that, we solve the Schrödinger equation. Here’s a primer on that too:
This gets fairly messy so I suppose you need not understand what I’m about to show you but what I do hope you take away from this is that if we wanted, we could solve this equation to find the behavior of our electron.
For a one-dimensional system like ours, the equation governs what the wave function looks like this:
- Mass, m
- Wave function, Ψ
- Position, x
- Potential, V
- Energy, E
We also see the reduced Planck’s constant, hbar, in the numerator of the first fraction. Once again, you don’t need to understand the details of this equation, the concept of quantum tunneling remains the same regardless. All you need to know is that it’s possible to solve this equation.
Anyway, the system proposed earlier can then be broken down into two scenarios:
- Case 1: the electron has more energy than the barrier → E > u
- Case 2: the electron has less energy than the barrier → E < u
Case 1 is fairly boring. All it tells us is that with more energy than the peak potential barrier, the electron can pass through. But we didn't need quantum mechanics to tell us that. We already knew this when we took classical physics into consideration. We don’t really learn a lot.
The more interesting case is where the electron has less energy than the potential barrier’s peak. E < u. In classical physics, we would've expected the electron to bounce back. As we’ve already shown, the probability of finding our electron on the other side would be zero. Or at least, so we thought.
In quantum mechanics, we’re considering the wave function of our electron. And the wave function behaves like a wave.
And when this wave function is solved for in the Schrodinger equation, it surprisingly yields a non-zero solution past the barrier. This means that we can find the electron in x > a region. But how would that be possible if the electron does not have enough energy? Well, the potential barrier, as far as we’re concerned, still fulfills its purpose as a barrier. What does happen, however, is that only a part of the wave function is allowed to pass through: the evanescent wave.
In quantum mechanics, evanescent waves are associated with tunneling phenomena, where a wave collides with a potential barrier, and emerges from the opposite side with reduced amplitude but the same energy as the incoming wave packet
A better visualization, perhaps, would be this:
Consider the y-axis as the barrier. As the wave approaches, it experiences a repulsive force but passes through with reduced amplitude. And when the amplitude is squared, we get the probability distribution. A non-zero amplitude means a non-zero probability. From this, we can conclude that particles can pass energy barriers even without sufficient energy.
This is quantum tunneling. Particles do have a chance at getting to the other side of the barrier even though classically, they didn’t have enough energy. Specifically, within the potential barrier, our wave function looks like this:
In the region where E < V, the curve is an exponential decay. And that’s exactly what we’d physically expect. The probability of finding a particle as we move further and further to the right is exponentially decaying. This means that the wider the barrier, the less chance we have of finding our particle on the other side.
What we’ve established here is that particles can move through barriers even though their energy is less than that of the peak potential. And with all that being said, I will end this article here. Thank you for reading!
