Category Archives: Physics

The Physics of Free Will

bluePillRedPillThe French mathematician and physicist Pierre-Simon Laplace famously boasted that if he knew the exact position and velocity of every particle in the universe at a given instant in time, he could predict with perfect precision the state of the universe at any time in the future (and, presumably, the past as well). Such was his faith in the universality, immutability, and sovereignty of the classical laws of motion, that he believed no particle, out to the very farthest reaches of space and back to the earliest moment in time, could escape the path predetermined for it by all of the interactions it was destined to have with the rest of the universe since creation. It is a nice-sounding boast, as it paints a picture of an unshakable, perfectly ordered world; and yet, it is terribly disturbing, as it utterly disallows any concept of free will whatsoever.

In Laplace’s world, the innermost workings of the human mind, down to our apparent ability to make decisions and move according to our will, are in fact governed by the laws of motion as they apply to the tiny particles that make up our brains and the physical processes that constitute thoughts — enormously complex to be sure, but entirely predictable with the right amount of knowledge. If true, Laplace’s boast would seem to be an end not just to free will, but to much of the meaning we find in life. After all, what significance is there in a work of art if the artist’s hands were merely being moved by the inevitable firings of neurons determined by the laws of physics since the beginning of time? Creativity would be only an illusion. There would be no spontaneity of thought or expression, no hope of controlling one’s own fate.

The performance of a symphony, with the musicians playing in harmony under the direction of a conductor and to the enjoyment of their audience, would in fact be nothing more than a fully predetermined combination of motions. The composition itself could no longer be truly ascribed to the composer, as he was predetermined from the beginning of time to write down precisely the notes of the piece being performed; and the husband who falls asleep in the audience could not be faulted for his inattentiveness, because that’s just how things had to play out. Even the guy whose cell phone rings during the adagio would be blameless.

But twentieth century physics has shown Laplace’s view to be wrong. We now know that pure chance plays a fundamental role in the outcome of any process. Quantum mechanics has shown irrefutably that particles actually don’t even have precise locations — it’s not just that we can’t measure them precisely enough to know where they are exactly, but rather there is no exact value to be measured. Nor do they have precise velocities. The highly celebrated but oft-misinterpreted uncertainty principle of Heisenberg describes not just limitations in our knowledge about the position and momentum of a particle, but the fuzziness of the particle’s actual being.

This is a huge leap in thinking that most physics students have trouble making, but it is at the heart of quantum mechanics. And not only are every particle’s actual position and momentum fuzzy in a fundamental way, but there is a very real element of pure chance involved in the particle’s behavior. An electron can disappear from one region of space and reappear in another region without crossing the space in between; and just where it reappears is a matter of chance that even the particle itself cannot know ahead of time. Einstein hated these revelations of quantum mechanics, but he recognized their truth. And so now we know on the basis of science alone — even if our bodies and souls are nothing more than extremely complex physical systems — that our fates are not entirely predetermined by the laws of physics. Certain outcomes are highly likely, to be sure, but never certain beyond all doubt (unless, of course, the “many worlds interpretation” of quantum mechanics is correct, in which case every possibility will, with certainty, come to pass in some universe).

So what does this new understanding of our world imply? Does it restore the free will and meaning that Laplace would have robbed from us? At first, it would seem that the answer must be no. The random events described by quantum mechanics cannot directly result in free will, for if they did, there must be a means for an agent of will to tell electrons (for example) where they should appear and interact with other particles (which is essentially the main physical process in our brains that is relevant to thought); but if this were the case, then the electrons would no longer be obeying the laws of chance that they have, in fact, been shown to obey. So it seems that random processes alone cannot account for free will.

However, our universe is governed not by chance alone, but by a most intriguing combination of deterministic rules and random processes — a continuum that fades from pure randomness at the infinitesimally small scale to pure determinism at the infinitely large scale. Could it be that the combination of these two components allows for the construction of something that amounts to more than the sum of its parts? In plane geometry, a straightedge limits its user to the construction of line segments, and a compass limits its user to the construction of arcs; but when the two tools are used in tandem, a whole new level of complexity becomes possible, allowing the geometer to draw impressive figures. Perhaps the classical, deterministic laws of motion and the more recently discovered quantum mechanical laws of chance are, respectively, the straightedge and compass that, when used together, allow for the construction of high-level phenomena such as consciousness and free will, which would otherwise be inaccessible under classical laws or the laws of chance alone.

If so, we would expect such phenomena to emerge in systems that exist at the boundary between the macroscopic scale, where deterministic laws prevail, and the microscopic scale, where quantum mechanics prevails. And it so happens that the human brain (and any mammalian brain, for that matter) is just such a system. The brain as a whole is a macroscopic system composed of networks that are just at the boundary between macroscopic and microscopic, which are further composed of individual neurons that belong to the microscopic realm. Could there be a more suitable system for emergent phenomena such as consciousness and free will to develop? (Some might suggest that the answer to this question is yes: a computer.)

The point of these speculations is not to demean humanity by reducing the soul to a mere physical construct — the above meanderings certainly prove nothing of the kind, nor are they intended to do so — but to suggest that there might be hidden potential in the physical substance of our universe. Matter, space, and time, together with the rules that govern their interaction, may contain some life and magic that we haven’t yet imagined. In Genesis 1:24, God says, “Let the land produce living creatures” [emphasis mine]. Is it possible that the land itself — the material substance of the universe — has, buried deep within it, the very components not only of life and consciousness, but free will as well?

Admiral Tarkin’s Split Second of Embarrassment

Tarkin_DS

Just look how embarrassed he is!

In this post, I’m adding an interesting extension to a popular physics problem that’s been circulating on the internet.

The original problem is to calculate the average power output of the Death Star when it blows up Alderaan, making the following assumptions:

  • The shot takes half a second.
  • Alderaan has the same mass and radius as the Earth.
  • The amount of energy delivered is simply equal to the gravitational binding energy of the planet (i.e., the amount of energy required to break it up into tiny pieces and separate them by huge distances).

The extension that I’d like to add is this: Calculate the recoil velocity of the Death Star, and discuss the effect that the associated acceleration would have on the occupants.

First, a quick recap of the original problem.

Gravitational binding energy of a planet is given by the formula

GravitationalBindingEnergy

where M and R are the mass and radius of the planet, and G is the gravitational constant. Plugging in the numbers for Earth gives us a total energy of 2×10^32 J. If this much energy is delivered in half a second, the power output required is 4×10^32 W. This is one million times the total power output of the Sun!

Wow.

Okay, now for the extension. The amount of momentum carried by light is simply the amount of energy in the light divided by the speed of light:

PhotonMomentum

By conservation of momentum, this must also be the amount of momentum carried by the Death Star as it recoils backward (ignoring certain subtle effects that won’t make a big difference in the result). To get the recoil velocity, we simply divide the momentum by the Death Star’s mass.

Crap. What is the Death Star’s mass? It’s got to be somewhere in that technical readout that R2 was carrying…

We’ll just have to get a decent approximation. According to Wikipedia, the diameter of the Death Star is 160 km, so it has a radius of 80 km. Assuming a density similar to that of our moon (3000 kg/m^3), the mass of the Death Star is 6×10^18 kg. In reality, since the Death Star isn’t solid, its mass would probably be less than this, in which case the actual recoil velocity would be greater than the number we’re about to get. But, hey, we’re just looking for an estimate anyway.

Okay, we now have everything we need to calculate the recoil velocity. Are you ready?

It turns out that the Death Star would have to recoil with a velocity of one hundred thousand meters per second. That’s about 360,000 km/h, or 200,000 mph, which is roughly eight times the Earth’s escape velocity. Accelerating to this velocity in half a second amounts to about 20,000 g’s. That’s more than enough to splatter all of the Imperial personnel on board, plumbers and all, into a runny pulp against the walls.

Just imagine the embarrassment on Admiral Tarkin’s face when he realizes, during that split second before he meets his demise against the wall of his command room, how stupid it is to stand on a battle station that’s about to put out so much energy in such a short time in only one direction!

Introduction to Acousto-optic Modulators

AOMpic

Hello! If you need to learn how to use an acousto-optic modulator (AOM), you’ve come to the right place. Click on the picture (above) or on the link below to download the PDF version of my beginner’s guide to AOMs.

A Beginner’s Guide to Acousto-optic Modulators

For my research, I recently had to learn how to use an AOM, which is basically a vibrating crystal that can be used to deflect or otherwise change the properties of a laser beam. I was unable to find a good one-stop source that explains in simple terms all the basics about how acousto-optic modulators work and how to operate an acousto-optic modulator along with all the necessary accessories, so I decided to make my own guide and share it. If you’re completely new to AOMs, you should find it helpful. It’s for total beginners. And if you need to make a presentation about AOMs, feel free to use material from this one. (Please give me credit, though.)

For best results, view it using the “fit to page” scale. That way, the pseudo-animations that you get by flipping the pages will work. If you find an error, please leave a comment below and I’ll try to fix it.

Thanks to Brian DeSalvo for helping me to understand AOMs. Any inaccuracies in the above document are due to me, not him.

Homage to Einstein

Einstein

I’ve decided that reading biographies is the best way to learn history. It’s also the best way to make up for the fact that you only have one life to live. I recently read Einstein: His Life and Universe, by Walter Isaacson. Here are my thoughts — more on the man than on the book.

Einstein really was head and shoulders above most other physicists. Most are specialists within a particular subfield, with the really good ones venturing into other subfields every now and then. Einstein dominated them all.

He essentially proved the existence of atoms with his paper on Brownian motion. He developed the statistical tools for describing the behavior of large systems of particles. With his work on the photoelectric effect, he lay the groundwork for quantum mechanics (and he was later involved in its further development). He shattered our traditional concepts of space and time with the theory of special relativity, and then he shattered them again with general relativity.

The remarkable thing is that Einstein did most of his work alone by using pure logic and performing “thought experiments” (gedanken). Or was he alone? At the patent office in Bern, where he had his “miracle year,” he had been reviewing dozens of patent applications that dealt with methods of synchronizing clocks separated by great distances with the use of light or electronic signals. Einstein himself said that he probably would not have had his breakthroughs if he hadn’t been working there; so at least his ideas were stimulated by the inventions of others, and in that sense he was not working alone.

Einstein’s life was filled with great irony and paradox. He married an older woman against his parents’ will; years later, when his son Hans Albert was going down the same path, Einstein protested just as his own parents had. He initiated a revolution in scientific thought and showed contempt for older scientists who were unwilling to accept the new understanding of nature even though the revolution was founded on their own discoveries; but then he himself was unable to accept quantum mechanics, which was founded on his own discoveries.

He was in many ways despicable. He essentially abandoned his first daughter and covered up her existence. The way he treated his wives was abominable. As I read about his family life, I found myself reviling him. But he became quite endearing after the age of about 50.

Among physicists, perhaps only Newton rivals Einstein.* Most people know of Newton’s “three laws.” But they probably don’t know that Newton discovered many other laws as well. Like Einstein, Newton had an unimaginably intense desire to understand in a systematic way how the world around him worked. It’s that drive, I think, that has been behind every major revolution in human history, for better or for worse.

* Gleick, James. Isaac Newton.

A Simple Introduction to Special Relativity

einstein2

Think Einstein’s theory of Special Relativity is beyond you? Think again.

It sounds intimidating. But what most people don’t realize is that it only requires a knowledge of the Pythagorean theorem, the definition of velocity (or really just speed, in the sense of distance divided by time), some basic algebra, and a willingness to embrace an unintuitive new understanding of time (and distance).

Here’s a simple introduction that I wrote for my 9th and 10th grade students at Village High School in May of 2010:

Relativity

Feel free to share it, copy it, distribute it, shred it, or burn it.

Okay, I admit that there’s actually a lot more to Special Relativity than what’s discussed in this little paper. The first three inescapable conclusions that emerge from the theory are that (1) time slows down in a moving reference frame (“time dilation”); (2) moving objects are shortened (“length contraction”); and (3) events that are simultaneous in one reference frame occur at different times in other reference frames. What’s hard to wrap your mind around is that these effects are not just matters of perception. Rather, the times and lengths actually change.

Those three effects are only the beginning, though. From them can be derived all sorts of other fascinating phenomena. Velocities, energies, momenta, and forces also change from one reference frame to another. The most amazing thing about Special Relativity, in my opinion, is the fact that the magnetic force is actually just a consequence of these time and length transformations.* One could say that the magnetic force isn’t even a real force. It’s an effect that arises as a consequence of relativity whenever an electric charge moves. (That’s why the “electromagnetic force” is considered to be just one force.) In fact, you can set up a situation in which, from the point of view of one person who’s sitting still, there’s a magnetic field; but from the point of view of another person who’s moving in a certain way, there is no magnetic field. It all depends on your point of view.

It’s tempting to go one step further and draw parallels between the physical theory of relativity and various philosophical ideas — relativism in ethics, culture, and religion, for example. It seems like there’s a scientific basis for saying that ideas that are right from one point of view might be wrong in another, and vice versa. But in fact, such thinking is contrary to the very heart of special relativity. The physical theory is built on the following two axioms: (1) The speed of light is always exactly the same in any reference frame; and (2) The laws of physics are always exactly the same in any reference frame. Thus, special relativity is actually a theory of absolutes. In fact, Einstein himself wanted to call it “Invariance Theory.”** It was other people who gave it the name “Relativity.”

* Haskell, Richard. “Special Relativity and Maxwell’s Equations.”

** Isaacson, Walter. Einstein: His Life and Universe.

An Interesting Physics Problem: The Airport Walkway

airportWalkway

In December 2008, one of my favorite students emailed me an interesting physics problem:

Suppose you’re in an airport and you need to get from point A to point B. For part of the way, there’s a moving walkway; the rest of the way, there isn’t. You walk at your normal speed, both on the moving walkway and on the floor. The questions are:

1. If you have to stop to tie your shoe, should you do it on the floor or on the walkway? (Your goal is to make the trip in as short a time as possible.)

2. If you can run for a short, fixed period of time, do you save more time by running on the walkway or on the floor?

3. How do the answers to (1) and (2) change if you take special relativity into account?

I wrote up a solution, which I’m very proud of. Here it is:

Airport Walkway Problem