Quantum Tunneling at Platform 9¾
an insight into quantum tunneling with Harry Potter
Have you ever wondered how Harry Potter passed or phased through the brick wall of Platform 9¾? Or have you imagined how The Flash, Vision, and Kitty Pryde phased through stuff? Well, today we are going to discuss all of these weird phenomena via Quantum Physics, more explicitly Quantum Tunneling. Speaking of Harry Potter’s tunneling, for non-Potterheads, here is a cool introduction from the Harry Potter wiki:
King’s Cross Station is considered one of the main train stations to serve London, England. Students of Hogwarts School of Witchcraft and Wizardry take the Hogwarts Express at Platform 9¾ every 1 September at 11 AM sharp. To get to Platform 9¾, one must run straight at the wall and tunnel between platforms 9 and 10. [1]
The Harry Potter world is explicitly magic. Now, without any magic would you have any hope of getting through? Throughout this article, we will build upon a fairly good intuition into this phenomenon using the notion and understanding of Quantum Tunneling. But first, let’s take a deep dive into waves since quantum mechanics is all about both particles and mostly waves.
Sound waves are pressure waves in the air. When Harry screams, he forces air out through his mouth and sets up a vibration that travels through the air in all directions. Whereas, light is a different kind of wave, an oscillating electric and magnetic field that travels through space — even the emptiness of outer space, which is why we can see distant stars and galaxies.
What happens when waves encounter an obstacle?
The apparent bending of waves around corners is an example of diffraction, which is a characteristic behavior of waves encountering an obstacle.
When a wave hits a barrier with an opening, the waves that pass through the opening don’t just keep going straight, they spread out in all directions. The wave’s wavelength and the size of the hole through which it travels determine how quickly it spreads. If the opening is much larger than the wavelength, there will be very little bending, but if the opening is comparable to the wavelength, the waves will fan out over the full available range.
If sound waves encounter an obstacle like a brick wall, poll, or a tree, they will diffract around it, provided the object is not too much larger than the wavelength. [2]
Result of Diffraction — Casting Shadows:
Light waves, on the other hand, have a very short wavelength — less than a thousandth of a millimeter. A hundred wavelengths of visible light will fit in the thickness of Harry’s hair. When light waves encounter everyday obstacles, they hardly bend at all, so solid objects cast dark shadows. A tiny bit of diffraction occurs right at the edge of the object, which is why the shadows of edges are always fuzzy, but for the most part, light travels in a straight line, with no visible diffraction.
Wait! Does that mean, there is a teeny tiny probability that Harry can pass around both sides of the wall particles at the same time? How’s this possible?
Well, it’s kind of absurd and can’t happen in real life. Let me explain this using an analogy of the Plank’s constant.
Plank’s constant analogy:
The wavelength of a material object is given by Planck’s constant divided by the momentum, which is mass multiplied by velocity. Planck’s constant, just like the mass of an electron, is a teeny tiny number — about 10³⁰ kilograms, or 0.000000000000000000000000000001 kg. Imagine an electron is moving at the speed of six million meters per second. It will roughly have a wavelength of about a tenth of a nanometer (0.0000000001 m) or so. The wavelength of an 80-pound (about 35 kg) human, like little Harry, is about 10³⁵ meters (0.00000000000000000000000000000000001 m), or a millionth of a billionth of a billionth of the wavelength of our imaginary electron. How does that compare to the size of two atoms of a brick wall? Well, Harry’s wavelength compared to the distance between two atoms is like the distance between two atoms compared to the diameter of the solar system. There’s no chance of seeing the wave associated with Harry diffract off a photon of a brick wall atom, let alone pass around both sides of the wall particles at the same time. In summary,
Long wavelengths diffract and short wavelengths vice-versa.
According to the equation above, if Harry has more mass, then he will have a teeny tiny wavelength, since his mass is inversely proportional to his wavelength. So, Harry’s wavelength doesn’t diffract. Therefore, Harry can not go to the left and right side of the wall particles at the same time at the platform nine and three quarters.
Is this how Quantum Tunneling actually works?
Well, for starters, possibly yes! But if you want to explore more and don’t want to end up like the guy in this [funny-video] at King’s Cross station, London, here is a thorough explanation of Quantum Tunneling via the analogy of Energy, followed by some real-world applications which we will discuss later in this reading.
Energy Analogy
Remember the equation of a moving object which has a kinetic energy of KE Jules?
Suppose we are throwing a ball upward at a considerable velocity. An instant after the toss, the ball is moving. So it has lots of kinetic energy, but it’s near the ground and has no potential energy. The total energy is thus equal to the kinetic energy. As the ball moves upwards, its kinetic energy decreases (because it’s not moving as fast as before), and its potential energy increases (because it’s higher off the ground). The kinetic energy level drops, replaced by potential energy, but the total energy remains the same.
As you can see in the picture above, at the peak of its flight, the ball has maximum potential energy, but no kinetic energy, because for a split second, it’s not moving at all. On the way back down, it goes through the same process in reverse: it starts with potential energy but no kinetic energy and ends up with maximum kinetic energy (the same amount it started with) but no potential energy.
The Forbidden Region / Barrier:
The ball can’t go higher than a certain height because that would need an increase in its overall energy, which isn’t possible. The maximum height the ball can reach with a given amount of energy is called the “turning point,” because the ball reverses direction at that point. Heights above the turning point are “forbidden,” because the ball doesn’t have enough energy to reach them. We will call this region “a forbidden region or a barrier”.
Uncertainty:
The Schrödinger equation uses the potential energy of a quantum object to predict what will happen to the wavefunction of that object, so every calculation done in quantum mechanics is fundamentally about energy. Take a look at the aforementioned equation as follows,
We know that the momentum determines the wavelength. Near the ground, where the kinetic energy is high, the ball should have high momentum and thus the wavefunction should have a short wavelength. Higher up, where the ball is moving slowly, the ball has low momentum, and the wavefunction should have a longer wavelength. We also expect the probability of finding the ball above the turning point to be zero, because the ball should never go higher than allowed by its initial energy.
To make a quick change in the wavefunction, a large number of wavefunctions with different wavelengths must be added together. With so many wavelengths, there is a lot of uncertainty in momentum, and thus a lot of uncertainty in kinetic energy. But we don’t have a large uncertainty in the kinetic energy since we know how hard we threw the ball.
With a small energy uncertainty, the position of the turning point is also uncertain, resulting in no sharp changes in the wavefunction and a wavefunction that extends into the forbidden region. The ball can’t both have a well-defined energy and turning point exactly where classical physics says it should. If we want a small uncertainty in the energy, we have to accept more uncertainty in the position, and it means that
“there’s a teeny tiny chance you’ll be able to find the ball at a higher position than classical physics allows.”
Let us apply this analogy and understand the fact that an electron can tunnel through a metal,
Inside of a solid object, for example metal, is a forbidden region. The potential energy for one object inside another is enormous due to interactions between the atoms that make up the objects.
The simplest example of this is an electron hitting a thin piece of metal, where the potential energy is much higher. When it reaches the edge of the metal, If the electron’s kinetic energy is large, it can convert most of its energy to potential, and still have kinetic energy left to move through the metal. If the kinetic energy outside is less than the potential inside the metal, though, there’s no way the electron can enter without increasing its total energy. The edge of the metal becomes a turning point, and the metal is a forbidden region: an electron coming in from the left bounces off the surface and goes back where it came from. An electron coming in from the right bounces off the other surface in the same way.
For a very narrow barrier, though, there is some probability of finding the electrons at the opposite edge of the forbidden region from where they entered. Beyond that point, they’re no longer forbidden to be there — they’re back out in empty space, and move off with the same energy they had at the start. Somebody watching the experiment would see a teeny tiny fraction of the incoming particles — one in a million, say — simply pass through the barrier as if it weren’t even there. This is called tunneling, because the electrons have passed the forbidden region even though it is impossible for them to be inside it.
The smaller height of the wave to the right of the barrier indicates that the probability of finding the electron on the right is much lower than the probability of finding it on the left.
The probability of tunneling “P(x)” decreases exponentially as the barrier thickness “x” increases
If you double the thickness, the probability is much less than half of the original probability. On the other hand, as the energy of the incoming electrons increases, they penetrate farther into the forbidden region, and the probability of one making it all the way through increases.
Like promised, let’s figure out how Quantum tunneling is used in real-life applications,
The Scanning Tunneling Microscope [3]
A scanning tunneling microscope (STM) uses electron tunneling to make images of objects as small as a single atom. The STM was invented in 1981 at IBM Zurich and has become an essential tool for people studying the atomic structure of solid materials.
An STM consists of a sample of electrically conducting material and a very sharp metal tip brought within a few nanometers of the surface of the sample. The tip is held at a slightly different voltage than the sample, so electrons in the tip want to move from the tip into the sample. The electrons can’t flow directly from the tip into the sample, though, because the small gap between the tip and the sample acts as a barrier preventing the movement of electrons. If the gap between the tip and the sample is small enough, though — a nanometer or so — there’s some chance that electrons will tunnel from the tip to the sample. That produces a small current, which can be measured. The tunneling probability (and thus the current) increases dramatically as the tip gets closer to the surface, so changes in the current can be used to detect tiny changes in the distance between the two — changes smaller than the diameter of a single atom. If you take a large number of height measurements at points on a grid, you can put them together to create an image of the individual atoms making up the surface of your sample.
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My favorite applications of Quantum Tunneling,
- reaction in the sun: https://youtu.be/lQapfUcf4Do
- tunneling in your home using a water mug: https://youtu.be/kvSlaIwUCuk
The videos that encouraged me to write this blog,
- Because Science https://youtu.be/8vBJyFPbv-A
- Because Science https://youtu.be/PXWYpRzlgQM
The following Youtube videos on Quantum Tunneling are from my favorite quantum physics enthusiasts,
- Veritasium https://youtu.be/WIyTZDHuarQ
- Because Science https://youtu.be/8vBJyFPbv-A
- PBS Space Time https://youtu.be/-IfmgyXs7z8
- Up and Atom https://youtu.be/WPZLRtyvEqo
- Physics Videos by Eugene Khutoryansky https://youtu.be/RF7dDt3tVmI
- minutephysics https://youtu.be/cTodS8hkSDg
- professor Sabine Hossenfelder https://youtu.be/BJA_faoIFco
Optional reading on Quantum Tunneling,
- https://www.scientificamerican.com/article/quantum-tunneling-is-not-instantaneous-physicists-show/
- https://www.quantamagazine.org/quantum-tunnel-shows-particles-can-break-the-speed-of-light-20201020/
- https://www.sciencedirect.com/science/article/abs/pii/S0370157398000222
- https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.70.77
- https://iopscience.iop.org/article/10.1088/1126-6708/2008/06/095/meta
