Visualizing Quantum Mechanics

Or how I learned to stop worrying and love the computer. [Note: There's a neat video below the fold. ]

A Confession

I was recently rereading the Feynman Lectures on Physics. If you haven't read them lately, I highly recommend them. Feynman is always a pleasure to read. As usual, I was surprised. This time the surprise came in lecture 9, which the way the course was laid out meant that this was something like the last lecture in the third week that these students had ever received of university level physics. The lecture is on Newton's laws of dynamics. The start is of course Newton's first (second) law, $$ F = \frac{d }{dt } (mv ) $$ which, provided the mass is constant takes the more familiar form $$ F = ma $$ After discussing the meaning of the equation and how in general it can give you a set of equations to solve, he naturally uses an example to illustrate the kinds of problems you can solve. What system does he choose to use as the first illustration of a dynamical system? The Solar System. That's right. Let that settle for a second. The sad thing is that if you caught me off guard before I read the lecture, caught me in an honest moment and asked me how you would solve the solar system, I would probably have launched into a discussion of the N-body problem and how there is no closed form solution to newtonian gravity that involves 3 or more bodies. (Depending on who you are, I might have then mentioned the recent caveat, namely that there is a closed form solution to the N-body problem, but that it involves a very very very slowly convergent series). Now, how can Feynman use the Solar System as his first example of solving Newtonian dynamics and I have told you that it's impossible as my first words on the subject? Well, the answer of course is that Feynman was much smarter than I am. Perhaps another way to say it is that in a lot of ways Feynman was a more contemporary physicist than I am.

A Realization

Physics education has changed very little in the last 50 years or so. Now in some ways this isn't a problem. The laws of nature also haven't changed in the last 50 years. What's unfortunate is that the tools available to physicists to answer their questions have changed remarkably. Namely, computers. Computers are great. They permeate daily life nowadays. They are capable of performing millions of computations per second. This is great for physics. You see, a lot of the time, as you all know, the way you achieve answers to specific questions about the evolution of a system is to do a lot of computation. So what did physicists do before computers? Well, a lot of time they would have to do a lot of calculations out by hand, but no one enjoys that, so a lot of times you would have to make sacrifices, make assumptions that meant that your analytical investigations were simple enough to yield tiddy little equations. This is reflected in the kinds of problems we still solve in our physics classes. I never solved the solar system in my mechanics class. I never did it because there isn't a closed form analytical solution to the solar system. But you know what... that doesn't matter. It doesn't matter in the least. Because while there doesn't exist a closed form solution to the problem, it is very easy to come up with a numerical approximation scheme (see Euler Method). You see, the point of physics is to get answers to questions. And the fact of the matter is that those answers don't have to be 'exact', they don't have to be perfect. They need to be good enough that we can't tell the difference between them being 'exact' and them being an approximation. To do this numerically with a pad of paper and a pencil is a heroic task. Do do this with a computer takes a couple of lines of python code and a couple seconds.

An example

As an example of the neat things you can do with a few lines of python code and a few minutes on your hand, check this out. and there's more This video depicts time dependent quantum mechanics. I set up a gaussian wavepacket, inside of a potential that includes a hard wall on the sides and is proportional to x. That sounds fancy but what it means is that this is the quantum mechanics equivalent of a bouncing ball. The amplitude of the wave function corresponds to the probability of finding the particle at any location. That is, imagine picking one of the colored pixels at random. If you pick any of the colored pixels at random, and look down at the x position, that is what measuring the position of the particle would do. But what are the colors? Quantum mechanical wave functions are complex. This means you can represent them either with a real and imaginary part, or with a magnitude and a phase. Here it's the latter. Like I said the amplitude is shown with the height (actually the amplitude squared). The color corresponds to the phase, where the phase is mapped to a location on the color wheel, just like the one that pops up in Photoshop or GIMP. And theres sound too! The sound is what the wave function would sound like if it was making noise. Its the real part of the wave function played as a sound. To that end, in this video it is very low frequency, because I made the movie slow enough to see the colors changing well. Its fun to watch the video and listen to the sound. For this movie the sound correlates nicely to when the 'ball' reaches its maximum height. Whats also cool is that you can hear the 'ball' delocalize after each bounce. The sound and function start off being nice and sharp, but after a few bounces it starts to spread out. You can also see how momentum is encoded in quantum mechanics. Funny thing is that instead of being something separate that you need to specify like in classical mechanics, in quantum mechanics the wave function is a complete description of the evolution of the system. I.e. if I showed you just one frame of this bouncing ball, you would be able to recreate the entire movie. If I showed you just one frame of a classical basketball, you'd have no idea what frame came next since you'd only know its position, not its velocity. In quantum mechanics the momentum gets encoded in the wave function, and as you can tell its encoded as a complex twist. A phase gradient. A crazy rainbow. If you look closely, you can even see that you can tell the difference between whether the particle is falling left or right. When it goes left the rainbow pattern goes (reading left to right) blue red green. When its moving right it goes blue green red. It twists one way then the other in the complex plane. The colors are a little hard to see in this one, they're a little easier to see in this one: This second one I dressed up a bit, labelling the axes with units, putting a time counter, superimposing the potential I was talking about, and marking the average expected position with a tracer black dot on the bottom.

A Call to Arms

Any student who has taken a first course in quantum mechanics knows enough physics to make these movies. The physics isn't complicated. But the movies really neat, right? More than neat. Making these videos taught me things about quantum mechanics I should have learned a long time ago. I really think computers are underestimated in physics education. They can be a great tool. A picture is worth a thousand words, so a movie must be worth millions. : denotes stolen quote More than just as an illustrative tool, the fact that even students in the first introductory mechanics physics course can solve for something like the solar system shouldn't be hidden from them. Classical mechanics after all is the physics of pretty much every object we can see and touch, but classics mechanics classes only ever talk about Atwood machines and frictionless planes. Often the closest they come to realism is in discussing projectile motion, where the laws you learn in the book (neglecting air resistance) are very good at describing the trajectories of very dense large objects (i.e. cannonballs). I can't remember the last time I've fired a cannon. But air resistance serves little trouble to my computer. Or Rhett's (of Dot Physics, which has just moved to Wired). Basically, if you give a student an intro physics course and an intro programming course, suddenly you have a human being who is better equipped to answer questions about natural phenomenon than 99% of human beings that have ever lived. So lets take a tip from Feynman and teach physics students how to solve the solar system.

Code

As per request, here is the python code I used to generate the videos. Its rather messy, so I apologize in advance. schrod.py - A general script which finds the eigenvalues and eigenbasis for a 1D particle with an arbitrary potential. qmsolver-bouncy.py - Code to generate the movie. You need to create a directory with the same name as the name in the script in the save folder as the script. The last two lines make the sound and the directory full of images. I used ffmpeg to wrap the two together.