Like Chords of Music: Quantum Tunneling

jonah

The world is a dynamic mess
Of jiggling things
It’s hard to believe
~Richard Feynman

The essential nature of matter
Lies not in objects, but in interconnections
Like chords of music, it’s beautiful
~Sophia Hoffman

Subway Tunnel
While we need tunnels to go through solid objects, quantum particles only probably need tunnels. The key word is probably (source).

+Dripto Biswas recently asked me through google plus to explain why a superfluid climbs up the walls of its container. I don’t know very much about superfluids themselves. However, I can explain the basic quantum mechanics behind their behavior. (Spoiler alert: I’m going to mention quantum tunneling!) It might be helpful to reveiw some of my previous posts on quantum mechanics. The most relevant ones will be my three part series on QM, and my post on the Pauli exclusion principle.

Before we go on, though, let’s take a look at this very cool video Dripto shared with me:

Excited? Great!

The Classical Turning Point

To better understand the bizarre nature of quantum mechanics, let’s first try to understand how classical Newtonian mechanics works. Imagine that Bart Simpson drops with his skateboard into an infinitely tall halfpipe. It doesn’t really matter what height he drops from, but we have to give it a name. So we’ll call the height h.

Actually, it's a simple harmonic oscillator.
A real half pipe (left) and our infinitely tall half pipe (right). The red line represents the height h at which Bart Simpson enters the half pipe.

When Bart enters the pipe, the Earth pulls at him and wants him at ground level. This is Bart’s potential energy. It’s equal to

    \[PE = mgh,\]

where m is Bart’s mass, and g is the acceleration due to the Earth’s gravity. Because Bart isn’t moving, his kinetic energy (the energy you get from moving) is zero. After Bart rides down the pipe so that he’s at ground level, all his potential energy has been converted into potential energy. Now Bart has no potential energy, but his kinetic energy is

    \[KE = mgh.\]

This is because energy is conserved.

Bart at the bottom of the parabola
Bart at the bottom of the half pipe. Now all of Bart’s potential energy has been converted into potential energy.

Incidentally, we can use this to calculate Bart’s speed. The definition of kinetic energy is

    \[KE = \frac{1}{2}mv^2,\]

where v is Bart’s velocity. So we can set the two definitions of kinetic energy equal to each other to find that

    \[mgh = \frac{1}{2}mv^2.\]

And then we solve for v:

    \[v = \sqrt{2gh}.\]

Now Bart’s kinetic energy is enough to bring him up the other side of the half pipe. But, he can only go so far. As Bart climbs, his kinetic energy is converted into potential energy as the Earth begins to pull on him again. Bart loses all his kinetic energy exactly after he has clmbed back up to a height of h.

Bart has reached the classical turning point!
By the time Bart reaches his original height, he’s run out of kinetic energy.

At this point, Bart will turn around and descend down the half pipe again. The astute among you will notice that this is a never-ending cycle: Bart goes down the half pipe and then rises up the other side to a height of h, and goes back down the half pipe. The points of the half pipe that are at height h are called the classical turning points, because these are the places where Bart has to turn around.

The classical turning points
The classical turning points are the places on the half pipe that are at the height Bart entered the pipe from.

Bart spends a lot of his time in the center of the half pipe: at the bottom. However, because he slows down as he goes up the sides of the pipe, he also spends a lot of time up on each side. To see this, we can imagine taking many many pictures of Bart skateboarding. Then, later, we shuffle all the still images so that we don’t know which picture comes after any other picture. Then we count the pictures where Bart is in any given position.

The figure below shows the probability of Bart being in any given location on the half pipe. Now the red line is both h and represents a zero probability of Bart being in a given position. Note that there’s some probability of Bart being anywhere in the half pipe. However, of course, it’s impossible for Bart to be outside the half pipe.

Probability distribution of Classical Bart
The probability of Bart being in any given location on the half pipe. Now the red line is both Bart’s initial height and represents a zero probability of Bart being in a given position.

The quantum case is very different.

A Quantum Bart

Now let’s consider the same situation. But, let’s imagine that Bart is a quantum mechanical particle. Now, Bart’s a wave. The height of the wave (roughly) represents the probability of finding Bart at a given position. The plot of Quantum Bart’s probability wave looks something like this:

The plot of Quantum Bart on the half pipe
The plot of Qauntum Bart on the half pipe.

At first, Quantum Bart looks pretty similar to Classical Bart. But, there’s some key differences. First of all, there are points where the probability of observing Bart is zero. In the classical case, Bart could be found everywhere with some probability. Second, Bart has a nonzero probability of being discovered outside the classical turning points! There’s some probability of him being higher up the walls of the half pip! (Indeed, there’s some probability of finding him being outside the half pipe!)

What’s going on? Well, to solve for Quantum Bart’s wave function, we need to solve the Schrodinger wave equation:

    \[\frac{\partial}{\partial t} \psi(x,t)= -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x,t) + (PE)\psi(x,t).\]

Don’t worry too much about understanding the equation. What I want to emphasize is that the equation is a wave equation, and it depends continuously on energy. What this means is that as the energy cost goes up, the probability of a particle being found at that energy gets smaller. However,  the probability is not zero. It only goes to zero when the energy cost is infinitely high. It’s very hard to understand the Schrodinger equation without a strong intuition for the behavior of waves, but that’s the idea.

The take home message, then, is that the position of a wave is not limited to where a particle can classically go. Because of this, quantum particles aren’t restricted by the classical turning points.

Bart Tunnels

We’ve seen some very weird behavior out of Quantum Bart so far. However, we haven’t seen Bart quantum tunnel. The issue is that there’s nowhere for Bart to tunnel to. Instead of the infinitely tall half pipe, now let’s look at a different situation. Let’s imagine that Bart’s standing in a room with no ceiling and the walls are some thickness. Classical Bart doesn’t have the kinetic energy to leap over the walls, so he’s stuck within the room.

Classical Bart stuck in a room
Classical Bart stuck in a room. h is the height that Bart can jump given his kinetic energy.

What about Quantum Bart? Quantum Bart is a wave, and we know that there are wave solutions for the regions forbidden to Bart, including inside the walls. Quantum Bart’s probability distribution looks something like this:

 

Quantum Bart in a box. It rhymes!
Quantum Bart in a box. Now, because there’s zero energy cost to stay outside the room, there’s a much bigger probability of Bart being in a classically forbidden region.

Now, the probability of Bart being inside the walls decays rapidly, and this makes the probability of Bart being outside shrink. However, there’s a pretty large probability of Bart being observed outside! So… what happens to Bart’s wavefunction when we observe him? Does he stay in the same state?

When we observe Bart, the place we observe him in becomes our new starting position and we adjust his wave function accordingly. So, if we observed Bart outside the room, his probability distribution would very suddenly change to the figure below:

Bart tunneled out
If we observe Bart outside the left wall of the room, then his probability distribution suddenly changes so he’s more likely to be outside.

In other words, Bart appears to have teleported outside the wall. This is quantum tunneling!

Okay, So Superfluids

(A special thanks to Matthew Lawson for helping me with the superfluidity theory)

So what does all this have to do with superfluids? A superfluid is a macroscopic quantum state (also called a Schrodinger cat state). In a macroscopic quantum state, we force a material to behave quantum mechanically on scales we can see, even though quantum mechanics usually only describes very small objects.

For a superfluid, we cool the liquid to as close to absolute zero as possible. As we do, the wave function of each particle in the liquid expands until the particle began to form overlapping waves. If the particles in the liquid are fermions, the Pauli exclusion principle forces the particle-waves to repel each other and nothing too weird happens. However, if the particles are bosons, they’re all forced to enter the lowest energy quantum state. When they all suddenly drop into this state, their wavefunctions overlap and they form a single quantum wave. This is why a superfluid flows with zero viscosity: it’s behaving as a single quantum particle!

This is also why a superfluid can climb the walls of its container. The fluid is climbing up the container the same way that Quantum Bart climbed up the walls of his infinite half-pipe past the classically forbidden points. In fact, if we’re very lucky, we could see a superfluid tunnel!

Important note: Really, superfluid helium behaves like a mixture of normal liquid helium and superfluid helium; and indeed the fraction of each can be measured. In principle a pure state would be achieved only at absolute zero (to satisfy the third law of thermodynamics)

Tunneling in Your Gadgets

Quantum tunneling isn’t just relevant for exotic states of matter. The scanning tunneling electron microscope uses this effect too. To measure the shape of a surface to very high resolution, it applies a voltage to the surface and the opposite voltage to a short piece of wire. The scientist then moves the wire over the surface at a very small fixed distance. Electrons tunnel from the material into the tip of the wire and create a current. The probability of an electron tunneling depends on the distance from the tip of wire to the surface. The shorter the distance, the bigger the tunneling current. We cna measure very short distances this way!

Scanning tunneling electron microscope
A diagram of a scanning tunneling electron microscope. (From Wikipedia.)

Further Reading

Here’s what I could find:

Questions? Comments? Insults?

I’m not an expert on superfluids, so if I got something wrong, and you know it, please correct me! And, as always, if you have any questions, comments, or personal attacks, please don’t hesitate to say so in the comments!

SOME CORRECTIONS

Earlier, I claimed that LEDs and transistors use quantum tunneling. This is false. I’ll describe how they work in a later article. However, I’ve already written about the underlying principle: band theory. Thanks for pointing this out, +Matthew Lawson.

On a similar note, my picture of Classical Bart’s probability distribution is wrong. I took the quantum solution, finagled it a little bit, and used it as the classical solution. Unfortunately, I didn’t actually calculate the solution like I did for Quantum Bart, and this doesn’t work. Bart’s probability distribution roughly mirrors the shape of the half pipe. This has been fixed. Thanks to +David Moore for catching this.

Sorry about these mistakes, guys. I wrote this article on 5 hours sleep at the airport and I finished at the last call for boarding for my flight home. It was a bit rushed, which is why I didn’t catch these mistakes.