# Quantum particle near an event horizon

As the next episode in our series about the Unruh effect (it gets hot when you accelerate), here can watch a video I have prepared which depicts how a quantum particle behaves near an event horizon.

So, what are we watching? The left and right panels show the spin-down and spin-up wavefunctions for a massless Dirac particle (a massless electron), initially at rest in Rindler spacetime. Colors correspond to phase. Because of the principle of equivalence, there are two alternate physical interpretations:

• You are moving with constant acceleration rightwards. At time t=0 you drop a Dirac particle. It seems to move leftwards, just because you leave it behind. When it gets far away, it slows down. This is due to relativistic time-dilation.
• There is a uniform gravitational field pointing leftwards. That’s why the Dirac particle accelerates in that direction. As it falls, it slows down. This is due to gravitational redshift.

Of course, the interference pattern which develops at the center is just quantum mechanics, nothing else. But when the particle reaches the edges of the box (top, bottom and right), new interference patterns appear which are spureous to our problem. That’s just the handicap of a finite-size simulation.

Nice, ein? This was work we developed at ICFO, Barcelona, along with Maciej Lewenstein, Alessio Celi and Jarek Korbicz. I have just showed it as a premiere during the Quantum gases meeting at CSIC in Madrid.

# Rough is beautiful (sometimes)

No posts for three weeks… you know, we’ve been revolting in Spain, and there are times in which one has to care for politics. But physics is a jealous lover… :)

So, we have published a paper on kinetic roughening. What does it mean? OK, imagine that, while your mind is roaming through some the intricacies of a physics problem, the corner of your napkin falls into your coffee cup. You see how the liquid climbs up, and the interface which separates the dry and wet parts of the napkin becomes rough. Other examples: surface gowth, biological growth (also tumors), ice growing on your window, a forest fire propagating… Rough interfaces appear in many different contexts.

We have developed a model for those phenomena, and simulated it on a computer. Basically, the interface at any point is a curve. It grows always in the normal direction, and the growth rate is random. The growth, also, is faster in the concavities, and slower in the convex regions. After a while, the interfaces develop fractal morphology. I will show you a couple of videos, one in which the interface starts out flat, and another one in which it starts as a circle. The first looks more like the flames of hell, the second more like a tumor.

The fractal properties of those interfaces are very interesting… but also a bit hard to explain, so I promise to come back to them in a (near) future.

The work has been done with Silvia Santalla and Rodolfo Cuerno, from Universidad Carlos III de Madrid. Silvia has presented it at FisEs’11, in Barcelona, a couple of hours ago, so I got permission at last to upload the videos… ;) The paper is published in JSTAT and the ArXiv (free to read).

# Who needs doors, when I can tunnel?

Tunneling is one of the mysterious features of quantum mechanics, but there is a very nice way to visualize it. There is a simulation method in order to obtain the ground state of quantum systems, called path integral Monte Carlo. It is based, as its name suggests, on Feynman’s path integral approach to quantum mechanics, but I will not go deeper in that… The idea is the following: a particle becomes a set of many, many copies, beads or replicas. Which one is the real particle? All of them, and none. Then, link them all, each one to the next, with a spring whose natural length is zero, and with a spring constant which increases with the mass. Now, put all the system in a “fake temperature”, which depends on $\hbar$… and that’s all! Simulate that, just using Monte Carlo, and the equilibrium distribution that you obtain is the ground state of the quantum system.

In the simulation, the potential is represented with the background colors: blue is low, orange is high. So, you see the potential consists of many minima, the central one is deeper. In fact, the energy of the particle is not enough to jump over the barriers… but it does not matter now. The “ring polymer” can jump, even if a classical particle can’t. The height of the barrier is not a huge problem if it is thin, because in that case the “spring” can stretch, and make one of the beads jump over it! That’s the tunneling, indeed.

So, what you’re observing in that video is the quantum cloud. In fact, each ring polymer represents a possible history for the particle, returning to the initial point.  Each bead corresponds to the position of the particle at a certain instant, and the energy in the springs corresponds to… the kinetic energy, which will not be zero because of the uncertainty principle.

If you need more explanations (I would!), read qfluct

# Quantum dreams

Quantum mechanics, the dreams stuff is made of… (David Moser)

A quantum particle, prisoner in a square box of infinite walls, starts out with minimal energy, which grows and grows, slowly… although, no matter how much energy it gathers, no matter it grows quadratically… it will never escape…

You can also see it as the vibrational modes of a square drum. It looks continuous because I interpolated between them for a smoother visualization…

# Sqpirals

This time the challenge is directly… to find out what’s that! I promise, it’s an extremely simple algorithm. I got the idea in a coffee-talk conversation, as good physics napkins should :), I don’t know if somebody has given them a name. For me, they’re sqpirals…

Njoy…

BTW, the final frame is, I know, my desktop… But I was too lazy to repeat the video! XD

# Three-body tango

Let’s start a new section in physics napkins, called scitoys, for scientific toys. The idea goes as follows: I write down code to illustrate something in physics or maths in an animation. The animation is displayed here, with some explanations and some ideas about further development… So, if you’re a jedi master, you can enjoy the challenges…

This first scitoy is just a running test… the ones I’m preparing are more spectacular. So, in three-body tango, we show three particles interacting through gravitational attraction. I’m pretty sure you all know by now that the two-body gravitational interaction (in Newtonian mechanics) is as regular as a cuckoo clock, while the intrusion of a third body makes it chaotic. You can see in the video how the planets change couple in a chaotic way…

Some technicalities… the program is in C++ for linux with X11. I will send the code to anyone interested and, in due time, I will publish it for all the scitoys. The video capture was done with xvidcap.

A technicality related to this particular program: the gravitational interaction has a short-distance cutoff. I mean: if the planets come closer than a certain minimal distance, I don’t allow the force to accumulate anymore. This avoids some instabilities…

And the challenge: how many perfectly periodic orbits can you find? Are they stable?

With a wink for Miguel Ibáñez Berganza & Daniel Gómez Lendínez