The Men Who Weighed Mountains

Title cover of discourse on the attraction of mountains
Figure 1. The original cover of the book “A Discourse on the Attraction of Mountains.” From Google Books.

 In 1687, Sir Isaac Newton published the Philosophiæ Naturalis Principia Mathematicahis magnum opus describing the laws of motion and the secrets of the universe. One such secret is Newton’s law of universal gravitation, which states that the same gravitational force that pulls us down to the Earth holds the planets in their orbits around the sun. Indeed, every mass attracts every other mass through gravity.

This means that not only are we pulled downwards towards the Earth, but we are pulled towards pieces of the Earth. We are all gravitationally attracted to mountains. In fact, this is an excellent test of Newton’s theory: if we could measure the gravitational attraction of a test mass to a mountain, we could confirm whether or not gravitation is indeed universal. And thus men began to weigh mountains.


Newton rejected the possibility of weighing mountains… he felt that the effect would be much too weak. However, others were convinced it should be possible. In 1738, French astronomers Pierre Bouguer and Charles Marie de La Condamine travelled to the mountain Chimborazo in Ecuador. They were there for other reasons, but they took the opportunity to test Newton’s theory. They brought a large, heavy pendulum with them, with a heavy mass suspended on a string. If the mass felt no force, other than the pull straight down due to the Earth, the string of the Pendulum should stand perpendicular to the ground. However, if the mountain attracted the mass, it should be deflected towards the mountain and the string wouldn’t quite be perpendicular to the ground, as shown in figure 2. Moreover, the amount the pendulum was deflected would be proportional to the ratio of the density of the mountain to the density of the Earth itself.

Schiehellion experiment
Figure 2. The forces on the mass at the end of a pendulum near a mountain. F is the attraction to the mountain, W is the pull of the Earth towards the ground, and T is the string holding the mass up. (Source: Wikimedia commons)

The conditions were difficult and the measurement wasn’t very precise. However, Bouguer and Condamine believed they had detected a deflection. They argued that this confirmed Newton’s theory. Moreover, they said, it showed that the Earth was not a hollow shell, disproving the belief held by several major thinkers of the day.


Given the tentative success of Bouguer and Condamine’s experiment, in 1772, Nevil Maskelyne, Astronomer Royal (that is a real title, I promise!) proposed a more careful repeat experiment. His proposal gathered quite a lot of enthusiasm and it became something of a political endeavor. The Royal Society of London formed a committee, the Committee of Attraction, whose members included Benjamin Franklin, to pick a mountain to use. And for political reasons, they wanted the mountain to be part of the United Kingdom.

Schiehallion viewed from Loch Rannoch
Figure 3. The mountain Schiehallion, where the second mountain attraction measurement was performed. Source.

With the aid of surveyor Charles Mason, the committee eventually settled on Schiehallion, shown in Figure 3 in Scotland. Then, with the aid of Charles Hutton and Reuben Burrow, Maskelyne performed the pendulum experiment. This time, thanks to the political enthusiasm, the experimenters had the time and money to do things right. They carefully surveyed the mountain and its terrain to measure any effects due to the curvature of the Earth and they purchased expensive surveying equipment.

The Maskelyne team’s endeavours were a complete success. After several years of work, they measured a deflection of the pendulum by 11.6 arc seconds. This told them that the density of the mountain was approximately half that of the Earth. By measuring carefully measuring the volume of the mountain and the density of the rocks composing it, they found that the density of the Earth was 4.5 times the density of water. This is different from the modern value by 20% or so! An amazing triumph.

Because of the Maskelyne team’s triumph, both Bouguer and Condamine’s and the Maskelyne team’s measurements are called the Schiehallion experiment.

One page of the Maskelyne team’s first report of their findings is shown in figure 4.

Figure 4. The first page of a report on the Schiehellion experiment. Source.

Related Reading

If you enjoyed reading this, you might enjoy reading about other historical experiments.

  • In this post, I describe the famous experiment that told us that the speed of light is constant.
  • In this post, I describe the photoelectric effect, which told us that particles are waves.
  • In this post, my good friend Michael Schmidt describes the Aharanov-Bohm effect.
  • In this post, I describe the experiment that discovered quantum spin.
Posted in History, Physics, Relativity | Tagged , , , | 2 Comments

Simulating Gamma Ray Bursts

A black hole eating a star.
Figure 1. Artist’s conception of a black hole eating a star. (Source, Wikipedia.)

It was the mid 1960s. The United States and the Soviet Union had recently signed the Partial Nuclear Test Ban Treaty, which forbid the detonation of nuclear weapons except underground. Since neither nation trusted the other, each was carefully monitoring the other for non-compliance. In particular, the United States feared that the soviets might be, I kid you not, testing bombs behind the moon.


The United States solved this problem with the Vela satellites. When a nuclear bomb goes off, it emits a short burst of gamma rays, which are rays of extremely high energy light. The Vela satellites were gamma ray detectors in space, orbiting the Earth 65,000 miles above the surface. Figure 2 shows one of these satellites in a clean room.

A Vela satellite
Figure 2. One of the Vela satellites in a clean room. (Source: NASA and Los Alamos)

The Vela satellites did detect gamma rays all right, but they didn’t come from nuclear weapons… they didn’t even come from the solar system. The satellites repeatedly detected short, very intense bursts of gamma radiation that nevertheless took too long to be from nuclear weapons blasts.

Gamma Ray Bursts

For a long time, we didn’t know anything about these events or what caused them. So we gave them the enigmatic name gamma ray bursts, and made up many models for what could cause them.

This changed in the late nineties, when we were able to measure X-rays and visible light emitted from the same source after the burst, which we call an afterglow. We now know that there are many causes for gamma ray bursts. Some bursts take a relatively long time and we’ve linked them to supernovae in distant galaxies.

The relatively shorter gamma ray bursts (creatively called short gamma ray bursts) are less common and less extensively studied. And we therefore know a lot less about them. One popular theory is that they’re caused by the merger of a black hole and a neutron star.

A Quick Aside on Neutron Stars

Neutron stars are the densest stars we know of. Figure 3 shows that a neutron star twice as massive as our sun might have a radius smaller than Manhattan. Indeed, the only thing preventing a neutron star from forming a black hole is the Pauli exclusion principle.

A neutron star superposed on Manhattan Island
Figure 3. A neutron star superposed on top of Manhattan island. In case you’re wondering, if this actually happened, the star would destroy the Earth. (Image source.)

Ordinary matter is made up of mostly empty space. The radius of an atomic nucleus is about a picometer, while the radius of an atom is about an angstrom. This means that, on average, 99.9999 % of matter is empty space. Not so with a neutron star. A neutron star is made up of neutrons packed as tightly as possible, like spheres. This means that in a neutron star, only about 25 % of a neutron star is empty space. (Obviously take this analogy with a grain of salt. The properties of a neutron star depend heavily on quantum mechanics and nuclear physics… so the neutrons aren’t actually packed like spheres. They’re waves.)

Anyway, neutron stars are incredible.

Black Hole-Neutron Star Mergers

When a black hole gets too close to a black hole, the black hole can eat it. But as I’ve discussed before, black holes are messy eaters. The matter in the neutron star gets distorted and forms an accretion disk around the black hole, which glows incredibly brightly.

As the accreting matter falls the black hole, that matter can be accelerated to incredible velocities and launched out the poles, forming an ultrarelativistic “jet.” These jets are common in many circumstances, but we believe that the jet from a black hole-neutron star merger might be the source of short gamma ray bursts.

There’s a long and beautiful history of studies of accretion disks and the jets they drive. And we’ve known that black-hole neutron star mergers produce accretion disks of the right type. But there’s still a lot we don’t understand about jets, accretion physics, and neutron stars.

The Jet Emerges: A Piece of the Puzzle

Recently, Vasileios Paschalidis, Milton Ruiz, and Stuart L. Shapiro, of the University of Illinois at Urbana-Champaign numerical relativity group, helped add a bit to our understanding. For the first time, they simulated a black hole-neutron star merger, watched as the accretion disk formed, and the relativistic jet emerged. This provides additional evidence that black-hole neutron star mergers might be the progenitors of short gamma ray bursts. Figure 4 shows snapshots of the simulation as the black hole disrupts the star, accretes the matter, and finally drives the jet.

a black hole eating a neutron star
Figure 4. Snapshots of the simulation as the black hole disrupts the neutron star, accretes the matter, and finally drives the jet. The white lines are the magnetic field lines, which drive the jet of matter. Image source.

Now, studies like this have been attempted before. Researchers routinely run simulations of black hole-neutron star mergers to make predictions about gravitational waves. And many groups around the world have run simulations of the jets driven by black holes. However, no previous simulation has successfully observed a jet after merger. All the previous jet simulations started with an accretion disk already in place.

Paschalidis, Ruiz, and Shapiro got their jet to emerge by correctly configuring the magnetic field of the neutron star before merger. Previously, all magnetic fields were assumed to be confined only within the star, and not exist outside it. Paschalidis, Ruiz, and Shapiro argue that this isn’t particularly realistic and, by including the exterior magnetic field, the jet emerges naturally.

This is a pretty cool piece of science!

Related Reading

If you found this interesting, you might enjoy my other posts on astrophysics.

Posted in Astrophysics, Physics, Relativity, Science And Math | Tagged , , , , , , , , , , | Leave a comment

Lightning Detection

Since I’ve been very busy lately my good friend Michael Schmidt agreed to do another guest post! Mike has a masters degree in physics from the University of Colorado at Boulder. You can check out Mike’s own blog at or his personal website Mike’s Personal Website. Without further ado, here’s Mike:

Lightning Detection


Currently, in the mid-west of the United States the first thunderstorms of the year have begun. Because I am a giant geek, I love lightning and I think tracking lightning is quite interesting. My personal favorite site is LightningMaps. On LightningMaps website you’ll see Google Maps overlaid with dots representing lightning strikes and circles emanating from them. The circles represent the leading edge of the thunder as it propagates away from the strike. Seeing this I immediately started to wonder how they do this and, appropriately, started to investigate.


The first step in any scientific endeavor is collecting data (Yes, yes. I Know. The scientific method starts with the generation of a hypothesis but in this case we know what lightning is and we simply want to monitor it). Lightning is a large surge of current, that is electrons, flowing between the Earth and the clouds above or vice-versa. Any time electrons are accelerated they emit photons which we usually see as visible light. In addition to the visible light we see, the lightning emits all sorts of other frequencies which include infrared (aka heat), ultra-violet, and radio waves. You can hear these radio frequencies if you happen to be listening to the radio when lightning strikes. They will sound like static. We can detect these radio waves and keep track of the time we received each burst of radio waves.


If we have only one radio station detecting these bursts of radio waves, we wouldn’t be able to tell where it came from since we would only be able to tell when we received the signal. Now, if there are two stations we can keep track of the difference in arrival times. We know light travels at a constant speed, c \approx 2.99 \times 10^8 \frac{m}{s} . If the first receiver picks up the lightning strike at t_1 and the second at t_2 we know the distance between the strike and the receiver 1 and the strike and receiver 2 is |t_1 - t_2| . This is known as the time difference of arrival or TDOA. Using this information we can only restrict the possible location of the strike to a hyperbola. A hyperbola is a conic section–a slice of a cone–which is the locus of points for which all points are a constant difference in distance between two fixed points or foci. This is exactly the case we have.

The form of any hyperbola is the following:

    \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

When plotted on a graph we see the following:

A Hyperbola
A hyperbola with two foci

We can fit this to the case we have where the two foci represent the locations of the stations and the time difference between arrival is fixed by setting a and b . This gives us both side of the hyperbola but only one side represents the possible locations. The otherside this the set of points where the TDOA is opposite that of what we observe. The plot looks like:

Possible strike locations if only two stations (red) are used. The yellow represents the actual lightning strike.
Possible strike locations if only two stations (red) are used. The yellow represents the actual lightning strike.

If we want to locate the position of the strike we will need a third station. With this third station we can create a second hyperbola which will intersect with the first exposing the location of the strike. In our example the second hyperbola has been drawn and it intersects at the location of the strike.

Two hyperbolas for two different pairs of stations showing the location of a strike.
Two hyperbolas for two different pairs of stations showing the location of a strike.

Solving the Equations

Unfortunately the equations used to describe these hyperbolas don’t lend themselves to a simple solution. There are solutions, for example Ryan Stansifer’s paper from Florida Institute of Technology, but they will fail if there is any error in your measurements. Instead, systems use a method called Gradient Descent. In gradient descent we create an error function which tells us how far off, in some way, from the solution we are and move in the direction which lessens that error. The the case of the lightning strike detector we know the time difference between two sites should be the measured TDOA so we can set the error for two stations to be

    \[err = [ (|\vec{x} - \vec{s}_1| - |\vec{x} - \vec{s}_2| ) - TDOA]^2.\]

Here (|\vec{x} - \vec{s}_1| means the distance between the position x and the first station s_1 . We square whole thing because we need positive and negative error to count against us. For our full error function we take every pair of stations and add up their error. What see what this looks like for our example above for the first pair of stations:

Error for two stations.
Error for two stations (red) for a lightning strike (Yellow)


We can see here there is a depression shaped like a hyperbola. This like the case with only two stations above only tells us the strike happened somewhere along that hyperbola.

If we add in the third point and account for its contribution to the error we get the following error plot:

With multiple stations the error function reveals the minimal error around the true location.
With multiple stations the error function reveals the minimal error around the true location.

Now we can find an approximate solution by letting a computer walk downhill and narrow in on the solution. To see this in action see this demo. At some point you tell the program to end when the error gets low enough. This is how sites like work where this process is repeated for every event they detect.

Hopefully this sheds light on this wonderful tool.

Further Reading

If you’re interested, here are some of Mike’s other guest posts:

  • Mike’s first two guest posts were on probability. In the first post, he describes how probability works. In the second, he covers some more advanced topics like probability distributions.
  • Mike also wrote a wonderful post on parallel computing.
  • Finally, Mike recently wrote a post on the Aharanov Bohm effect.
Posted in Electronics, Geometry, Mathematics, numerical analysis, Physics, Science And Math | Tagged , , , | Leave a comment

The Long Arms of the Black Hole

The Galaxy Centaurus A and the jets from its supermassive black hole.

Black holes are incredibly messy eaters. As matter falls into a spinning black hole, that matter can be accelerated to incredible velocities and launched out the poles. In the case of the supermassive black holes at the centers of galaxies, these are the most energetic events in the universe since the Big Bang.

The exact mechanism for the creation of these jets is unknown. There are two competing theories, one called the Blandford-Payne mechanism, and one called the Blandford-Znajek mechanism. The details are too fiddly to get into here, but the former has more to do with the in-falling matter and the latter has to do with how magnetic fields interact with the spinning black hole.

The image above is of the galaxy Centaurus A and the jets produced by its super-massive black hole, which is fifty five million times the mass of our sun. The white glow and brown disk are the galaxy itself and associated dust cloud respectively. The blue line is the ultrarelativistic jet of material emitted by the black hole. (Actually, it’s the X-rays emitted by the fast-moving matter in the jet.)

You can’t see the black hole at all. Even on the scale of a galaxy, it’s just a dot, smaller than a pixel. But it has a wide wide reach, extending far beyond the galaxy and influencing the growth and evolution of the galaxy profoundly.

(The image is actually the composite of three images. From Wikipedia: This is a composite of images obtained with three instruments, operating at very different wavelengths. The 870-micron submillimetre data, from LABOCA on APEX, are shown in orange. X-ray data from the Chandra X-ray Observatory are shown in blue. Visible light data from the Wide Field Imager (WFI) on the MPG/ESO 2.2 m telescope located at La Silla, Chile, show the background stars and the galaxy’s characteristic dust lane in close to “true colour”.)

Image Source:

Related Reading

These objects are called active galactic nulcei. See:

Black holes glow for other reasons too. They have so-called accretion disks, which glow incredibly brightly. I wrote about this a while back:

Posted in Astrophysics, Physics, Science And Math | Leave a comment

Aharonov-Bohm Effect

Since I was busy last week and I’m feeling ill this week, my good friend Michael Schmidt has agreed to write a guest post for me this week. Mike has a masters degree in physics from the University of Colorado, an interest in teaching, and a passion for math and physics.  You can find out more about him on his personal website or read more on his blog,

So, without further ado, here’s Mike’s article.

Force Vs. Energy

When we teach physics, usually force is one of the first concepts. Force is easy to understand. I can have you imagine riding in a car riding around a curved road. As the car accelerates, the seat pushes you along. When the car turns you can feel the seat push you in the direction of the curve. In fact, force is such an understandable notion we often neglect to ask what force is or if there may be a better way to talk about the world.

What is force?

Newton’s notion of force is the method which physically exchanges momentum. If two objects interact, they change each other’s momentum. Think of a two billiard balls bouncing off each other. If you placed your finger between between the balls you could feel a considerable force (don’t really do this, it will hurt). The billiard balls feel force due to the other and bounce off each other.

Now, this is how we speak about interactions for the most part. We draw force diagrams and use them to create equations we can solve. This, however, is not always so simple.

Let’s consider two similar examples where a ball bearing rolls (frictionlessly) down a slide: one where the slide is a straight slope and the second the slide is curved. Now suppose you want to find out how fast the ball will be moving when it gets to the bottom of the slide assuming it was nearly stopped at the top. In the first example at every point on the slide the effective force on the ball is constant. This is due to the slope being the same, what is true for one part is true for any other. Since the force is constant we can use the constant force equations to solve this.

Two similar cases of a ball bearing sliding down a slide, one a simple straight slide, and a more complicated slide second.
Two similar cases of a ball bearing sliding down a slide, one a simple straight slide, and a more complicated slide second.

Now, the second case. This situations is substantially more difficult, we need to recompute the force for every point along the slide.
We can’t use any convent equations we have to derive them. This can certainly be laborious and is not preferred.

Wondrously, there is a better way: energy.


Energy is a strange notion; unlike force, you can’t feel energy.
The rules of Newtonian mechanics can be used to create two quantities: kinetic energy (or KE) and potential energy (or PE). Energy, unlike force which has a strength and direction, is just a number. Kinetic energy, roughly, is how much work it takes to accelerate an object up to some speed, whereas potential energy is the capacity for an object to acquire kinetic energy. In other words, potential energy is energy that can become kinetic energy in the future.

Energy may flow between each of these types of energy but their total must always remain the same. To illustrate this imagine a spring fixed to a table on one end and let there be a weight on the other.

An oscillating weight on a spring. (From Wikipedia En:User:Svjo)
An oscillating weight on a spring. (From Wikipedia En:User:Svjo)

If you pull the spring to one side, stretching the spring, and release the weight it will move back and forth. When the weight is at the resting position of the spring, the weight will be under no force and will be traveling as fast as it can go, since as it continues to move it will be slowed again by the spring. It’s at this point the weight has it’s maximum kinetic energy and it’s minimum potential energy since the weight will not be sped up anymore. In contrast to this point, at both ends of the oscillation, the weight will stop. Here, we say the kinetic energy is zero and the potential energy is maximum.

If we use the notion of energy, we can make any situation like the bearing on the ramp nearly trivial to solve. This works since energy is allowed to be either kinetic or potential and the total must always be same. For the ball bearing on the slide example, the ball has only potential energy at the top of the slide and only kinetic energy at the bottom. We can represent this in an equation by

    \[PE + KE = C\]

    \[mgh + \frac{1}{2} m v^2 = C\]

Since C is an constant, you can make both sides equal for the beginning and end:

    \[mgh = \frac{1}{2} m v^2\]

We can then solve for the v and we would know the final speed of the ball. This method is has some obvious advantages, but all it seems we have done is find a quantity which hides the force.

What is Energy?

Potential and kinetic energy seems just to be abstractions of force. In other words, energy isn’t real, the force is. We just made up energy to make the math easier.

This certainly seems like the right answer, especially in the light of how we can actually feel force and energy can only be referred to in equations. Of course, I would not have said that if it were so simple. Quantum mechanics seemed to turn the scientific world on it’s head but could the notion of force be false too?

The Aharonov-Bohm Thought Experiment

Double Slit Waves/Interference
From Wikipedia’s en:User:Lacatosias and en:User:Stannered

This experiment begins with the double slit experiment, which shows the wave-particle duality of electrons. The double slit experiment has three elements to it: an electron emitter, a solid panel with two parallel cuts or slits in it, and a phosphorescent screen all arranged in this order. The setup is shown in the following image:

Double Slit Experimental Setup

As the particles move away from the emitter they pass through the slits and interact to create the interference pattern show here:

From Wikipedia's  en:User:Lacatosias and en:User:Stannered
From Wikipedia’s  en:User:Lacatosias and en:User:Stannered

The additional element added by the Aharonov-Bohm experiment is a very long solenoid encased within an impenetrable shell. The solenoid is place between the screens and the slits. A diagram for this is here:
Double Slit Setup (solenoid)

This solenoid will create a magnetic field inside itself but not outside. This means under our view of things that there ought to be no changes to the setup outside the solenoid as the magnetic field cannot possible be exerting forces on the electrons. Interestingly, as you change the magnetic field strength the interference pattern on the screen will move. This effect is named the Aharonov-Bohm Effect after its discoverers. How could this be though, there is no force on the electrons! In fact there is no magnetic field anywhere the electrons are. The answer is there is another field present, the vector-potential. The vector-potential is a way to abstract the notion of a magnetic field and it is non-zero outside the solenoid. If it were just a mathematical trick, we would say it being non-zero outside is a side-effect of the math and is inconsequential. However, as we see the strength of this field has a direct impact on our observable world.

This questions our assumption that force is the most primitive or basic of interactions. Perhaps our mathematical trick is the real thing. There is much debate about this and there is likely no simple answer. The notion of force isn’t useless but it does have it’s limits. Maybe at some point a future experiment will help out understand more. For now, we’re stuck without an easy answer.

Further Reading

If you would like to learn more about quantum mechanics Jonah has written a number of articles you might find interesting.

Jonah has a three-part series on quantum mechanics:

  • In the first part, he introduces particle-wave duality.
  • In the second part, he describes matter waves using the Bohr model of the atom.
  • In the third part, he describes how one should interpret matter waves.

Jonah also wrote a post describing quantum tunneling.

More recently, Jonah wrote a two part series exploring the relationship between particles and waves.

  • In the first part, he shows how you can build a particle up from a bunch of waves.
  • In the second part, he shows that this isn’t always possible.
Posted in Physics, Quantum Mechanics, Science And Math | Tagged , , , | 1 Comment

Spin and the Stern-Gerlach Experiment

Figure 1. A plaque at the University of Frankfurt commemorating the Stern-Gerlach Experiment. (Source)

The word “quantum” means a single share or portion. In quantum mechanics, this means that energy comes in discrete chunks, or quanta, rather than a continuous flow. But it also means that particles have other properties that are discrete in a way that’s deeply counterintuitive. Today I want to tell you about one such property, called spin, and the experiment that discovered it: the Stern-Gerlach experiment.

(The goal of the original experiment was actually to test something else. But it was revealed later, after the discovery of spin by Wolfgang Pauli, that this is in fact what Stern and Gerlach were measuring.)


The Stern-Gerlach experiment involves magnetic fields. So before I tell you about the experiment itself, I need to quickly review some of the properties of magnets.

As you probably remember, the north pole of a magnet is attracted to the south pole of other magnets and repelled from their north pole, and vice versa—a south pole is attracted to north poles and repelled by other south poles. In other words, opposites attract.

Suppose we generate a very strong magnetic field (say, with a very big magnet or with a solenoid) and put a small magnet in the field, as shown in Figure 2. What happens to it? The north pole of the big magnet will attract the south pole of the small magnet, and the south pole of the big magnet will attract the north pole of the small magnet. Since the north and south pole of the big magnet are are equally strong, these attractions will be equal and opposite, and they’ll cancel each other out so that the little magnet feels no net force. As a result, it doesn’t move up or down—it just hovers in place.

Figure 2. We put a little magnet in a big magnetic field and see what happens.

Now suppose we create a big magnet whose north pole is more powerful than its south pole, as shown in Figure 3. (It’s not actually possible to make a magnet with a stronger north pole than south pole. However, we can create the same effect by using multiple smaller magnets.) What happens now?

Figure 3. If we create a magnetic field where the north pole is much stronger than the south pole, the behavior is different.

To answer this question, we must understand that the strength of a magnetic force depends on the distance between the interacting poles; the closer the poles, the stronger the force. This means that the net force the little magnet feels depends on its orientation, as shown in Figure 4. If the south pole of the little magnet is close to the north pole of the big magnet, the little magnet will be pulled upwards. If, on the other hand, the north pole of the little magnet is close to the north pole of the big magnet, the little magnet will be pushed downwards. If the poles of the little magnet are the same distance from the poles of the big magnet, the little magnet will feel no force. And of course, anything in between is possible. A little magnet whose south pole is just barely closer to the big north pole will feel a weaker pull than a little magnet whose south pole is very close to the big north pole.

Figure 4. If we put a little magnet in a magnetic field that’s stronger near the north pole and weaker near the south pole, the force on the little magnet depends on its orientation.

The Stern-Gerlach Experiment

The Stern-Gerlach experiment, performed by Otto Stern and Walther Gerlach, tested whether subatomic particles behaved like little magnets. To do this, Stern and Gerlach created a magnet with a bigger north pole than south, just like the one described above, and shot a beam of electrons with random orientations through the resulting magnetic field. If electrons behaved like little magnets, then the beam would be spread out by the magnetic field, as shown in Figure 5. Some electrons would be pulled upwards, some would be pushed downwards, and some wouldn’t change direction, depending on the orientations of the individual electrons. But if electrons didn’t behave like magnets, then none of them would be affected by the magnetic field, so they would all just fly straight through.

Figure 5. What we expect to happen in the Stern-Gerlach experiment  if electrons behave like little magnets.

Surprisingly, although the electrons were affected by the magnet, they didn’t spread out as in Figure 5. Instead, the electrons split cleanly into two beams, as shown in Figure 6.

Figure 6. What actually happens when the Stern-Gerlach experiment is performed on electrons. The electron beam doesn’t spread out like a fan, but splits neatly in two!

That’s very weird! It implies that electrons behave like little magnets, but only sort of. A magnet can be oriented any way it likes. But an electron can only have two orientations: either aligned with the big magnet or aligned against it. So the electron can travel up or down, but it can’t stay in between. This is a distinctly quantum phenomenon—the electrons behave like magnets fixed into a pair of discrete orientations, or states, as opposed to a continuum of possible orientations. An electron’s spin is what describes which of those two states it’s in.


A Cool Video

Here‘s a cool video I found on Wikipedia that shows what I just explained.

Where Does Spin Come From?

I won’t discuss it in detail here, but we can understand spin as emerging from the structure of the underlying quantum field theory that describes the behavior of a given particle. For those of you who know the lingo, it has to do with whether the underlying field is a vector or scalar field, and how large that vector is. (Among other sources, see Quantum Field Theory in a Nutshell by Anthony Zee.)


The Stern-Gerlach experiment reveals a dramatic difference between the quantum world and the world we’re used to. It’s not possible for a particle to have any old orientation; it must be oriented either with the external magnetic field or against it.

But what if there is no external magnetic field? How is the particle oriented? Somehow the act of measuring the system changed how it behaves, or at least how we perceive it. These are questions that physicists struggled with in the early twentieth century as quantum mechanics was being discovered. Indeed, to some extent, physicists are still struggling with them.

In the next few weeks, I’ll address some of these issues. Next time, I will talk about an extension of the Stern-Gerlach experiment that helps us explore, if not answer, some of these questions.

Related Reading

This is only the latest in a number of articles that I’ve written about quantum mechanics. For example, I wrote a three-part introduction to the field:

  • In the first part, I describe some of the experiments that first revealed particle-wave duality.
  • In the second part, I use the Bohr Model of the atom to explain how packets of energy emerge from the wave nature of matter.
  • In the third part, I describe how we can interpret matter waves as probability waves.

More recently, I wrote a pair of posts exploring particle-wave duality.

I’ve also written a number of stand-alone articles on quantum mechanics:

  • Quantum mechanics uses complex numbers, so I wrote a short explanation of imaginary and complex numbers here.
  • I explain the Feynman path integral, which is a way of understanding quantum mechanics, here.
  • I use particle-wave duality and matter waves to explain quantum tunneling here.
  • I use quantum mechanics to describe how atoms form covalent bonds here.

Further Reading

Here are some additional resources on the Stern-Gerlach experment:


Thanks as always to Alexandra Fresch for her line-editing.

Recently I’ve had a lot of discussions on Google+  about the interpretation of quantum mechanics. (In particular, I’ve spent a lot of time talking to +Charles Filipponi and +David R.) This article was partly inspired by those conversations. Thanks, guys!

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The Universe Is an Inside-Out Star

the universe is an inside-out star!
The universe as an inside-out star. The stellar surface is the cosmic microwave background. Behind it is plasma. And behind that, the center of the star is the Big Bang. (Thanks to Douglas Scott for generously providing these images.)

No, not really. But as we’ll see, it’s a useful analogy. Today we’ll learn about sound waves in the sun and how, if we imagine that the universe is the sun but inside-out, these are the same as the sound waves that filled the early universe.

DISCLAIMER: This is a pedagogical exercise only! I am not claiming the universe is ACTUALLY an inside-out star or that scientists think of it as one.

Sound Waves in the Sun

I’m sure you won’t be surprised when I say that the sun is a complicated beast. A nuclear furnace burning at tens of millions of degrees powers a burning ball of turbulent hydrogen gas and plasma. All sorts of crazy things happen in the sun. Magnetic fields reconnect and plasma flows on the surface, neutrinos fly out of the nuclear reaction in the core, et cetera. But let’s ignore all that for now. Let’s say that the sun is “just” a gigantic ball of superheated hydrogen gas.

But hydrogen gas is… well, a gas. And if something makes a noise, sound can travel through it. Moreover, how the sound travels,  and the frequencies that make up the sound, can tell us a lot about the interior of the sun. Fortunately for us, lots of things in the sun make sound. For example, if a bit of gas is hotter than its surroundings, it will create a pressure wave through the sun. And this pressure wave is nothing more than a sound wave.

But if we want to use these sound waves to understand the interior of the sun, we have to measure them. How on Earth do we measure sound in the sun?

Wiggles Beget Wiggles

Fortunately, we don’t need to measure the sound waves directly. All we need to do is measure the color of the light coming off the surface of the sun. A sound wave is just a fluctuation in the velocity of the particles that make up a gas. So, as a sound wave reaches the surface of the sun (called the photosphere), it will accelerate the atoms in that area. This in turn slightly changes the color of the light these atoms emit, thanks to something called the Doppler effect. (I’ve spoken about the Doppler effect before in the context of the expanding universe.) Atoms moving toward us emit light that is more blue than it otherwise would be, while atoms moving away emit light that is more red. Since not all light coming from the sun is emitted at the surface, the change in the color of the sunlight that reaches us is small but measurable.

Therefore, all we have to do is look at the surface of the sun and measure the changes in the color of the light emitted from different points on the solar surface. These changes in color correspond to the peaks and troughs of a sound wave traveling through the sun. The scientific field that studies the sun’s interior using the color fluctuations on its surface goes by the awesome name of helioseismology.

The Universe

So what does all of this have to do with universe at large? Well, as I’ve remarked before, we know that the early universe was filled with an extremely hot plasma—so hot that atoms and molecules couldn’t form. And this plasma glowed incredibly brightly. As the universe expanded and cooled, atoms and molecules formed, but the glow remained. It still remains today in the form of a bath of microwave radiation filling the universe, which we call the cosmic microwave background, or CMB for short.

That’s one way to look at things. But there’s another way, too.

Looking Back in Time

The speed of light is finite. Indeed, it’s the speed limit of the universe. This means that the light from a star four lightyears away from us is four years old. In other words, when we look out into space, we look into the past. And greater distances take us further back in time.

As we peer away from Earth, things are mostly empty for a while. Stars and galaxies are incredibly far apart, after all. But eventually we peer far enough away, into the extreme past, that we see the hot plasma of the early universe. The plasma is opaque, though, so we can’t see inside it. What we can see is the point when the plasma cools enough for atoms to form. The distance at which we see this happen is called the surface of last scattering. The corresponding time in the history of the universe is called recombination.

Since we can’t see inside the plasma, it might seem impossible for us to learn what happened before recombination. But it’s plausible that the plasma fluctuated and moved… and maybe sound waves even traveled through it. Fortunately, we can measure that! The fluctuations in the pre-recombination plasma change the color of the light in the cosmic microwave background!

And now we’re at the punchline. One way to understand this is to imagine that the universe is an inside-out version of the sun, as shown in the figure. As we look away from the Earth, backwards in time, there’s empty space. Then we reach the surface of the universe-sun, which is nothing more than the surface of last scattering. Behind it is the plasma which makes up the interior of the universe-sun. The sound waves in the interior change how the atoms and molecules on the surface (the surface of last scattering) move and thus change the color of light that’s emitted and eventually reaches us!

And thus, by measuring the fluctuations in the CMB, we can measure the dynamics of the very early universe!

The Big Bang Wasn’t a Point

One thing I like about this analogy is that it takes the center of the sun, which is a single point, and smears it out so that it becomes the surface of a very large sphere, one with the same radius as the observable universe. I like this because it reverses a common misconception.

People usually imagine that the Big Bang, the beginning of the universe, was  a single point from which everything emerged. This is completely wrong. The beginning of the universe happened about fourteen billion years ago at every point in space. So, in our inside-out sun analogy, the smeared stellar center is the Big Bang.

(Of course, there may not have been a Big Bang if, for example, cosmic inflation is correct. But that’s a story for another time.)

Related Reading

What I described in this post is a weird and crazy way of looking at the cosmic microwave background. But I’ve discussed the more “standard” understanding of the CMB several times. Most recently, I described the nitty-gritty of how cosmologists measure the CMB and how this is related to the failed BICEP2 “primordial gravitational waves” measurement.

I also wrote a three-part series on the early universe:

  • In the first post, I describe how the cosmic microwave background helped convince scientists of the existence of the Big Bang.
  • In the second post, I describe some problems with the Big Bang theory.
  • Finally, in the third post, I describe how the model of cosmic inflation fixes the problems with the Big Bang.

Further Reading

  • This post is inspired by—and borrows heavily from—a pedagogical paper by Crowe, Moss, and Scott, which you can find for free here. It’s very readable, even for the layperson, so I recommend checking it out if you’re interested.
  • Astrophysicist Brian Koberlein has a beautiful (pun intended!) blog post on how we probe the interior of the sun, in which he describes helioseismology and some other techniques. You should definitely check it out.
  • There’s a nice piece in Scientific American on the CMB here.
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Tidbit: Radio Waves Bouncing Off of an F-15

A simulation of radio waves bouncing off of an F-15 fighter jet. Left: the fighter jet is constructed out of many tetrahedra, inside each of which lives many waves. Right: the electric field on the surface of the jet. (Source: Gottlieb and Hesthaven, Spectral Methods for Hyperbolic Problems)

I’m afraid I don’t have time to write very much this week. So instead, I leave you with a little hint of the sort of thing I’m thinking about. The above picture is from a paper I just read. It shows a simulation of radio waves bouncing off of an F-15 fighter jet. The simulation was effected by first building the jet out of many tiny pyramids linked together at the faces (shown on the left). Then, a set of five waves or so was allowed to exist inside each pyramid. When you take all of these waves together, you get the radio wave that’s hitting the jet (shown on the right).

I’m working on taking this technique and using it to simulate relativistic astrophysics, like black holes and supernovae.

I’ll have a lot more to say on this eventually, but for now back to work!

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The BICEP2 Result Was Just Dust, and That’s Okay

Planck on Earth
Figure 1. The Planck satellite being prepared for tests. (Image due to the Planck Collaboration and the European Space Agency.)

You may remember that about this time last year, there was a big hullabaloo because a research group from Harvard claimed that they had discovered primordial gravitational waves using BICEP2, their telescope in the South Pole. This was very exciting because, if true, the result would be extremely good evidence for a model of the early universe called cosmic inflation. (Cosmic inflation is mostly accepted by the scientific community, but it has some philosophical problems and is thus still a little bit controversial. The BICEP2 results would have ended the controversy once and for all.) Even better, the precise strength of the primordial gravitational waves measured by BICEP2 was much greater than anyone had expected, implying that there might be some exciting new things we could learn by studying them further.

Unfortunately, the BICEP2 team’s results were quickly called into question. A competing, space-based experiment, Planck, released some of their own data indicating that what the BICEP2 team had observed was simply cosmic dust. But the analysis wasn’t definitive. The jury was still out.

The beautiful thing about science, though, is that scientists collaborate. The BICEP2 team and the Planck team got together, shared data, and worked through a joint analysis of their measurements. This analysis took several months, but it’s finally been released.

Now we know definitively. The BICEP2 measurement was indeed cosmic dust, not primordial gravitational waves. But the jury is still out on the existence of primordial gravitational waves. It’s just that, if they exist—I personally think they probably do—then they’re as weak as (or weaker than) we originally thought, not as strong as the BICEP2 measurement indicated.

But why did BICEP2 get such a strong false positive? How did their measurement go so wrong? Well, hold on to your hats, ladies and gentlemen, because I’m going to explain.

This is a follow up to my post from last March on the BICEP2 result. If you haven’t read that yet, I suggest you go check it out before continuing.

A Much-Too-Short Summary of Cosmic Inflation and the CMB

About 13.8 billion years ago, the universe was extremely hot, so hot that matter couldn’t form at all… it was just a chaotic soup of charged particles. Hot things (and accelerating charges) glow. And this hot soup was glowing incredibly brightly. As time passed, the universe expanded and cooled, but this glow remained, bathing all of time and space in light.

(The reason for why the universe was so hot in the first place depends on whether cosmic inflation is true. Either it’s because the Big Bang just happened or it’s because, after cosmic inflation, a particle called the inflaton dumped all of its energy into creating hot matter.)

Even today, the glow remains, filling the universe. As the universe expanded, the glow dimmed and its light changed colors (due to gravitational redshift), until it became microwaves instead of visible or ultraviolet light. This ubiquitous glow is called the Cosmic Microwave Background, or CMB for short, and if you turn an old analogue TV to an unused channel, some of the static you hear is CMB radiation picked up by your TV antenna.

Since its discovery, the CMB has been one of our most powerful probes of cosmology. It lets us accurately measure how fast the universe is expanding, the relative amounts of normal stuff vs dark energy and dark matter, how the density of matter fluctuated in the early universe, how the Earth is moving relative to the expansion of the universe, and much more.

Measuring the CMB

One amazing thing about the CMB is that all of the light that reaches us is the same color, to an incredible degree. However, the color does fluctuate a little bit…in a special way that’s independent of position in the sky or direction. These tiny deviations from the norm are primarily what we’d like to measure.

So how does a measurement work? How can we measure something that’s literally everywhere? On Earth, we can see things in three dimensions because we have two eyes separated from each other. But on the scale of the CMB, which fills the entire universe, the whole Earth looks like a single eye–in other words, from our perspective, the sky is two-dimensional.

Figure 2. Although light from the CMB (red) fills the universe, when we observe it on Earth, it looks like it’s been projected onto a two-dimensional screen (blue) above our heads. (Earth courtesy of NASA.)

This means we observe all of the light from the CMB as if it were projected onto a spherical screen above our heads, as shown in Figure 2. Looking from the outside in, the result is something like Figure 3, which plots the wavelength of the light across the sky. (The differences in the wavelengths have been enhanced by about a factor of a million.)

the CMB on our sky
Figure 3. The cosmic microwave background projected onto our sky, looking from the outside in. (Image made by Damien P. George with data from Planck.)

Of course, although three-dimensional models are easiest to visualize, they’re not great to actually work with. So we usually map the CMB onto a flat surface, the same way we map the Earth. This is what gives rise to the famous “all-sky” maps like the one shown in Figure 4.

Figure 4. The measured CMB mapped on a flat surface. (Image due to the Planck collaboration.)

There’s a lot of information hidden in Figure 4 that you can’t see unless you do some serious math. In fact, you could learn almost everything I’ve told you so far just from looking at the CMB! And there’s more to learn as we make new and increasingly precise measurements.

Planck Vs. BICEP

It’s at this point that I need to provide a clarifying comparison. The images I just showed you were generated by the Planck satellite, which is a small satellite that lives just beyond the moon’s orbit. As Planck orbits the Earth (and as the Earth orbits the sun), it makes measurements of the CMB in small segments of the sky. Over the course of a year, it can build up a map of the CMB in the entire sky, as shown in Figures 3 and 4. (Planck also takes measurements of several different wavelengths of light and aggregates the data. This is important and I’ll get back to that.)

BICEP2 (shown in Figure 5),  on the other hand, is a single telescope near the South Pole. The BICEP2 people chose to measure a small patch of the sky extremely precisely and they only measured one wavelength of CMB light.

BICEP2 sunset
Figure 5. The BICEP2 telescope at the South Pole. (Courtesy of the BICEP collaboration).

What Went Wrong?

If sending a satellite into space or pointing a telescope at the sky were all that was required to precisely measure the CMB, the BICEP2 team never would have mistaken dust for gravitational waves. So what went wrong?

Well, I told you that the fluctuations in the CMB are very very small. This means that they can be drowned out by the many other sources of microwaves in the universe. Jupiter, the sun, black holes, pulsars, cosmic dust…tons of things produce microwaves. Collectively, all this other stuff is called foreground.

To screen out the foreground, cosmologists build an extremely detailed map of non-CMB sources of microwave radiation in they sky, called a mask, and subtract it from the map of microwave light that the instrument actually measured. After the subtraction, you get something like Figure 4. The mask used to remove known sources in the Milky Way is shown in Figure 6.

Planck Galaxy Mask (for the Power Spectrum)
The mask that the Planck collaboration used to screen out galactic microwave sources from their measurements of the CMB. (Source: the Planck collaboration)

But mask-making is tricky business. To build a map, cosmologists use previous measurements of the sky and computer simulations. The Planck collaboration uses an additional trick: they can detect several different wavelengths of microwaves. The only microwave source that will look the same in every wavelength is the CMB, so by comparing the measurements in different wavelengths, Planck can remove unexpected sources of noise.

But BICEP2 only measured one wavelength of light, and this is what killed it. The computer models the BICEP2 people used to make a mask for their little corner of sky didn’t predict that it contained as much spinning cosmic dust as it does. Planck, with their multi-wavelength detector, wasn’t fooled in the same way.

(I should emphasize that the BICEP2 team’s mask was flawed. The team based their dust estimates on older measurements and made a mistake when estimating how much the radiation from the dust would change when you looked at a different color of light. But these are subtle errors, and having several colors of light to look at would have been a fail-safe against them.)

BICEP2 Didn’t Do Anything Wrong

It’s tempting to say that the BICEP2 collaboration failed in some way—their data analysis was poor, they designed their experiment badly, etc. But they couldn’t have known that this cosmic dust would have been a problem. It’s easy to see what to do in hindsight…not so much when you’re planning a multimillion-dollar project years (or even decades) in advance.

This is how science is done. We make a prediction, we design an experiment, we measure something in the world, and we invariably mess up. But by keeping our minds open to our own fallibility, we give ourselves the opportunity to try again and eventually get it right. That’s what happened here. BICEP2 made an erroneous conclusion, took the opportunity to collaborate with Planck, and they figured it out.

That’s fantastic. That’s what I call science.

(Parenthetical note. There are other ways that the BICEP2 team deserves criticism. Before submitting their article to peer review, the team held a huge press conference and generated a lot of publicity. Given that the conclusion wasn’t yet vetted by the scientific community, this kind of behaviour can and probably did detract from the credibility of science in the public’s eye.)

Further Reading

  • If you’d like to read the joint Planck-BICEP2 press release, you can find it here. In the press release, there’s an link to the scientific paper that the two collaborations wrote together, which is currently undergoing peer review.
  • Planck periodically releases their measurements and statistical tools, including software, masks, and all-sky maps, to the scientific community at large. This is where I got the galaxy mask I showed. If you’d like to browse, you can find all the data and documentation explaining how to use it here.
  • If you’re very brave, you can read this review paper on how foregrounds are removed from CMB measurements. This article describes in great detail how masks are generated.
  • If you want to look at the CMB in its full spherical glory, Damien P. George created this webapp. It’s pretty awesome.

Related Reading

  • This article is a follow-up to my previous article on BICEP2, which you can find here.
  • If you’d like to know how we know that the universe is expanding, you might want to check out my article on exactly that.

If you’re confused about the Big Bang or this whole “inflation” thing, you might want to check out my three-part series on the early universe:


Thanks to Alexandra Fresch for proof-reading and editing and thanks to Sara Simon for making sure I get the cosmology right.

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Sometimes a Particle Isn’t Possible

That blob turns into two!
Figure 1. That blob splits into two! This animation shows what happens when we try to construct a single particle (with the given shape) and make it bounce between two mirrors.

Last time, I showed you how you could construct a photon, a light particle, in a configuration of mirrors called a ring cavity. This time I’ll show you that sometimes, you can’t make just one particle—they only come in pairs. And sometimes, the notion of a particle doesn’t make any sense at all. (This post relies heavily on last week’s post, so if you haven’t read that, I recommend you do so.)

Disclaimer: What I’m about to describe is only the simplest case, and I make simplifications for the sake of exposition. It is possible to capture and manipulate single photons between two mirrors for short times if you play tricks. In fact, that work recently won the Nobel prize.

Last time, I showed you what happens when you arrange three mirrors to make a ring. Now let’s see what happens when we bounce light between two parallel mirrors, as shown in Figure 2. This is called a Fabry-Perot cavity.

A Fabry-Perot Cavity
Figure 2. Light (yellow) bounces between two mirrors (light blue).

We’re going to put waves into our Fabry-Perot cavity and see if we can make just one particle. (Spoiler alert: it won’t quite be possible!)

As I’ve discussed before in some detail, light is an electromagnetic wave made up of electric and magnetic fields. To draw a parallel to our example last week, the strength of the electric field can very roughly be thought to correspond to the probability of measuring a photon. However, there are complications; for example, the quantum-mechanical wavefunction can be imaginary. It’s an experimental and theoretical fact that electric fields are zero inside conducting materials like metals. (This isn’t quite true…the field actually falls off slowly, based on something called the plasma frequency. But we’re making approximations.) Therefore, the electric field that makes up a photon must be zero at the metal mirrors.

This means that if we put a wave of light between the two mirrors and look at the strength of the electric field (which wiggles), it has to go to its center position at the mirrors, as shown in Figure 3. This restricts the type of wave that can fit in between the mirrors. (On our plot, the field is zero when it’s smack dab in the middle of the plot. Above the center line, it’s positive. Below the center line, it’s negative.)

Figure 3. If you put a light wave (red, green, blue) between two mirrors (dark blue), the wiggles have to end at the mirrors.

Last time, when we added a wave to our ring cavity, the wave travelled uniformly to the right with some speed. That’s not what happens now. Now the wave can’t travel, so the height just grows and shrinks. Let’s look at the longest possible wave that can fit in the cavity, shown in Figure 4. This is called a standing wave.

Fabry-Perot with one wave
Figure 4. Our Fabry-Perot cavity with the largest possible light wave sitting in it. The wave just grows and shrinks in time.

(There are complications, of course; I’m completely ignoring what the magnetic field is doing. But for explanatory purposes, this is enough.)

Now we can add additional waves to the cavity. If we add the first five that fit (in special amounts based on a mathematical calculation using Fourier analysis), we get a plot that looks something like Figure 5.

Figure 5. The first five largest waves in the Fabry-Perot cavity.

We seem to have some complicated wave motion here! Let’s add even more waves! If we add nine waves to the cavity, we get something like Figure 6.

Nine waves
Figure 6. Nine waves in a Fabry-Perot cavity.

If we add nineteen, we get something like Figure 7.

Figure 7. The Fabry-Perot cavity with 19 waves in it.

Now what’s happening is beginning to become clear. If we extrapolate to Figure 1, we see this:

We attempt to put a wave with a particle-like shape into our cavity, but it splits into two waves which fly apart, reflect off of the mirrors, pass through each other, and continue reflecting for all eternity.

In this case, it’s not possible to put just one particle in between the mirrors.

Sometimes Particles Just Don’t Make Sense

The example I’ve just described highlights a problem with the standard popular narrative of particle-wave duality. We’re told that particles sometimes act like particles and sometimes act like waves. But if this were true, a single particle would never split into two just because we dropped it between two mirrors. The truth of the matter is that everything is a wave. It’s just that sometimes, like in last week’s experiment, waves can be made to act like particles.

But this week’s experiment shows us that sometimes, waves can’t be made to act like particles–at least, not a single particle. And sometimes they refuse to behave like particles at all! What all of this means is that there are conditions where particles cannot exist. For example: We think that, about 13.8 billion years ago, the universe underwent a period of rapid inflation. During this expansion, for reasons that I promise to try to address in the future (see Mukhanov and Winitzky), the very notion of a particle broke down. In the inflationary period, the packets of waves that make up particles simply could not form.

Related Reading


I know I’ve been lazy about citing my sources on this blog and I should be better about it…even when the sources are not layperson-legible. So, for that reason, I offer the intrepid student a list of introductory texts he or she can use to learn more.

  • Introduction to Electrodynamics by Griffiths offers a comprehensive introduction to electromagnetic theory (e.g., how light behaves).
  • You can also find a simpler, more accessible introduction in the Feynman Lectures on Physics, which are available for free online.
  • A Fabry-Perot cavity can be approximated as a particle in an infinite square well. This problem, as well as particle wave duality and basic Fourier analysis, are all covered at an introductory level in the excellent text Modern Physics For Scientists and Engineers by Taylor, Zaphiratos, and Dubson.
  • A more advanced student may want to check out Introduction to Quantum Mechanics by Griffiths.
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