So I’m still mired in final exams–this time a final project for my quantum field theory course. The downside is that it will be yet another week before my next “real” post. The upside is that I still have a little something for you all this week. The above image shows part of what I’m working on for my project.
Imagine that you make a square box of mirrors, and with some magic quantum tweezers, you put exactly fifty-one photons into your box. Light is a special oscillation in an electromagnetic field, which we usually describe classically. But if we only have fifty-one photons…we need quantum mechanics. What do we do?
Quantum mechanics is a theory of probabilities, so we need to make the field behave probabilistically. What we do is enforce the Heisenberg uncertainty principle on our field and allow it–nay, force it to wobble randomly. That’s what my image shows; I modeled a massive scalar field, which is simpler than the electromagnetic field but an acceptable approximation for our purposes. The right plot shows the field before you throw in quantum mechanics. The left plot shows one measurement of the field after you throw in quantum mechanics. Each time we measure the field, it will look like some random fluctuation around the plot on the right.
Cool, huh? I promise I’ll have a real post on quantum field theory at some point.
Editors Note:This week, I’m busy with final exams here in Guelph, so my good friend Michael Schmidt has graciously agreed to do a guest post. Thanks, Mike!
Hi everyone! Since last time I decided to talk about the basics of probability, I thought this time I would expand on that subject. In part 1, I discussed how to count different possible outcomes of random events and determine the likelihood of particular events. If you have not read that, or it’s been a while, you should read over Part 1. This method is great when where are relatively few possible states but becomes burdensome when you introduce more complicated setups. As usual, when you try to model things like people and their actions there are many random factors which are not always easy to predict. Enter distributions, a wonderful tool for troublesome situations!
One of the most basic problems where the total number of states becomes troublesome is Galton’s Board. Galton’s Board (or the bean machine) is a panel with pegs, arranged in rows, a channel where balls or beans flow onto the topmost row of beans, and collection channels where the balls exit the rows of pegs and are stacked for counting.
The reason this is of interest is whenever a ball hits a peg it can go left or right. If the machine is built with enough care, the ball will have a 50% probability of going left and 50% probability of going to the right. As you can see in the following image, the ball will encounter a peg many times on the way down. At each peg the question of direction will be revisited at which time the ball may change directions.
Interestingly, there are many paths that lead to the same bin. This means we have to count up all the possible paths to figure out what the probability of finding the ball in a particular bin might be.
Let us look at the leftmost bin and see what paths end up there. It seems there is only one path, that is at each peg the ball goes left. If at any time it goes right, it will be unable to get back. Since there is only one path we multiply the probabilities of going left by each other and multiply it by one, the total number of paths. This gives us a probability of , which turns out to be in our example. The second bin on the left, however, has multiple paths. It requires the ball to go right only once at any point. Since there are pegs, the ball may go right once at any of those points; this gives us different paths. This means the total probability is which in our example is . The third bin from the left is a little more tricky. It turns out it requires two right bounces. This means there are different paths. Here, is the binomial coefficient or choose function. It gives the total number of ways you can choose objects from a total of . The algebraic form of the choose function is
This allows us to know how many total paths there could be. In the case above, so the total number of paths is . The probability is therefore which is in our example. This trend continues in the same fashion which gives us a general form for bin to be
If the machine were not built well there could be a bias to one side or another. To model this we can prescribe different probabilities to left or right action. In that case we get the following probability per bin:
where is the probability of the ball falling to the left and is the probability of the ball falling to the right. This is known as the binomial distribution.
What is a distribution? In short, it is a ways of laying out different bins or groups and prescribing probabilities to each of them. Most everyone is familiar with the bell curve; the bell curve is, as it turns out, a distribution. In math circles, the bell curve is usually referred to as the Normal Distribution. The normal distribution lets us model the results of many random trials which can interact with each other. This is usually the case for exam grades and the like. Each person taking the test has had a large number of different experience which have prepared them for the exam questions. Since this is the case you would expect the grades to fall along a distribution like that below:
You may now ask how this connects with the previous distribution. The answer is if we have a large number of rows of pegs then we will start to get curves that look more and more like the normal distribution. Below, I’ve included an animation of a binomial distribution when the number of pegs is increased.
In fact this trend to always begin to look like the normal distribution isn’t a coincidence but rather this will always happen when a large number of random data is taken. There are some conditions on that statement but I’ll leave that to those who are curious. This property is called the Central Limit Theorem. This fact means there is a lot we can learn about random events if we study the normal distribution.
Some Things About The Normal Distribution
The normal distribution is interesting as it’s mean and median are the same. That is the average value is also the value that splits the population into two even groups. This value is represented in the general equation below as . In addition, the width of the normal distribution is also characterized. This term is called the variance or standard deviation, . While these two have different strict mathematical definitions, you can think of this term as dialing in the width. Pictorially, this is represented by the following diagram:
and we can express the functional form as
The power of this distributional representation of possible outcomes is you can look at the sum of the little areas under the curve and get an approximation for the percentage of events within that range. (For experts, this is just the integral of the distribution between the two points.) For example, suppose you had test scores that fell along a distribution with a mean of 60% and a standard deviation of 10%, this would result in a curve like that below.
In this figure I’ve highlighted the range from 70% to 100%. This area represents 15.9% of the test-takers meaning we expect 15.9% of the total population to have scored above 70%.
How Distributions Apply To Physics
Distributions are particularly helpful to quantum physics as they can be used to describe where a particle might be found. Suppose a particle is trapped in a box, the particle’s position will be probabilistic, meaning it is not localized in any particular part of the box but rather, there are place where it is more likely to be found. I won’t now go into the details but it can be shown that a particle is, in its lowest energy state, distributed like so between two impassible barriers:
As you can notice, the most likely place for the particle to be found is in the middle of the box. In fact 50% of the time in with be found within the following highlighted area:
The second energy level is a bit more curious of a distribution, it looks like this:
The distributions in quantum mechanics will continue to behave even more interestingly as the setups get more complicated. However complicated they become, the methodology outlined here is the same. Probability underpins all of quantum mechanics and, hopefully, I’ve equipped you with a little more understanding.
It does seems strange the quantum world acts with such indeterminacy. This notion is certainly distressing as our macro experiences of the physical world are so predictable, however, it seems to stand the tests of science. Einstein famously disagreed with the idea that nature was intrinsically random by saying: “God does not play dice”. While we are not certain nature is random, our experiences lend credence to that effect. Since quantum mechanics has existed, it’s theories have been instrumental in our understanding of nature and it has lead to the creation of lasers, microscopes, computer hardware, and countless other technologies.
Jonah says:If you liked Mike’s post, you might also enjoy the articles I wrote about quantum mechanics.
This week I’m in Savannah Georgia for the April APS meeting. So far, it’s been a blast! I met fellow blogger +Hamilton Carter, who writes at Copasetic Flow. If you’re interested in relativity or the history of physics, you should definitely check it out. He had a very nice talk on the history of special relativity, and he blogged about it here. And next week, I’ll be taking an exam. So for the next two weeks there may be no posts.
Last week, I explained Rayleigh and Raman scattering from a classical point of view. In the process, I explained why the sky is blue and introduced Raman spectroscopy, a powerful tool for studying the structure of molecules. This week, I fill in the gaps and explain scattering from a quantum-mechanical point of view.
The story of energy levels starts deep within the atom. I’ve previously discussed the fact that particles are waves and how this means that electrons can only have certain specific energies inside an atom. The energy and momentum of a particle control how many times the corresponding wave wiggles within a certain distance. As shown below, these wiggles (wavelengths) must fit in a circle around the nucleus of the atom–the electron can’t cut off its oscillation halfway through to fit itself into an orbit!
If the atom is part of a molecule (especially a crystal), the discrete allowed energies become so numerous that, together, they look like a continuous band. And this leads to band structure.
For clarity, physicists often imagine extremely simple atoms with only two or three allowed electron orbits, each of which is allowed only at a single specific energy and a single specific momentum. Depending on the situation, they even neglect the momenta and only look at the allowed energies. This is what we’ll do. For example, the figure below shows a two-level atom with a single electron in the lowest energy state.
When a photon–a light particle–hits the atom (or, alternatively, passes right through it), it has the potential to affect the electron. If we ignore quantum mechanics and look at this classically, the light would always accelerate the electron, since the electron is a charged particle and electromagnetic fields affect charged particles.
However, if the electron accelerated, it would gain kinetic energy. This gain is only allowed if the electron ends up with one of the allowed energies–and if the electron is accelerated, it will absorb the photon’s energy and momentum. So it can only absorb the photon if the electron’s new energy and momentum are allowed within the atom. Otherwise, surprisingly, the photon passes right through the atom unmolested, as shown below.
Importantly, once an electron absorbs a photon, it can sit in the higher energy level as long it likes. It’s under no obligation whatsoever to drop down to a lower energy level state.
(Astute regular readers may complain here. In the past, I said that electrons want to be in the lowest-energy state available. Both statements are true. In the idealized situation, electrons stay in whatever energy state they’re currently in unless provoked by a photon. But in the real world, electrons drop to lower-energy states through fluorescence. This is because, in the real world–thanks to quantum field theory–there are always photons or other particles for the electron to interact with. And these other particles allow the electron to drop down to lower energy levels through stimulated emission.)
We cannot know both the position and the momentum of a particle at the same time. In other words, we can’t precisely know where a particle is and how fast it’s going at the same time.
And this is true, but it’s not useful for us. We’d rather restate the uncertainty principle equivalently as
or, in English,
Over short times, we cannot precisely know the energy of a particle. Only if we wait long enough can we accurately measure its energy.
The consequences of this restatement are a little nuts. Written this way, the uncertainty principle tells us that a particle can have enormous energy, so long as it has that energy for only a short time. Perhaps more importantly, if the electron is in an atom or molecule, that enormous energy doesn’t have to be quantum-mechanically allowed.
(I know the uncertainty principle seems crazy and unintuitive. There are a few helpful thought experiments that I’ll try to write about in the future.)
Virtual Energy Levels
This means that I fed you a little white lie earlier. I told you that an electron in an atom (or molecule) won’t absorb a photon if that photon has the wrong energy. But this isn’t quite true. The electron can absorb the photon, so long as it doesn’t keep the photon’s energy for long.
In this case, an electron jumps up into a so-called “virtual energy state,” which can only exist for short times thanks to the uncertainty principle. Then, before the uncertainty principle is violated, the electron emits a photon in a random direction, allowing it to drop back down to its original allowed energy state.
Because the energy difference between the virtual state and the original state is equal to the energy of the absorbed photon, the electron must vent precisely that amount of energy. So it emits a photon of the same energy–and thus color–as the original photon.
Wait…a photon is absorbed and then re-emitted in a random direction? Those of you who read last week’s post know that that’s Rayleigh scattering! Thus, this is the quantum-mechanical description of how light bounces off of an atom or molecule. (Important note: the description is a bit different for metals, which are reflective.)
Just to tie everything together with last week: In the classical picture, we treat a photon as a wiggling electromagnetic field, which causes our electron to wiggle in its orbit around the atom (or molecule). Although this uses up the photon’s energy, the wiggling electron then recreates the photon traveling in a random direction. In the quantum picture, the electron absorbs the photon, jumps up to a virtual energy level that’s allowed only by the uncertainty principle, then drops back down to its original energy level, emitting a new photon of the same color in a random direction.
But, as we discussed last week, things are a little different if light scatters off of a molecule. In an atom, electrons are localized to one nucleus. In a molecule, the electrons have several atoms to roam across. (As I discussed in my post on bonding, atoms in a molecule share electrons.) But atomic bonds in molecules are not static things. Because of the heat in the molecule, the atomic bonds wobble and vibrate all on their own.
This wobbling of the atomic bonds not only contributes to the kinetic energy of the electrons, but acts as an additional allowed energy level for the electron. So if we place our two-level atom (above) into a molecule, it will then have three levels (below): two atomic energy levels and one kinetic energy level that comes from the vibration of the molecule. Usually the vibrational state is at a much lower energy than the atomic excited states.
Now, when a particle absorbs a photon with the wrong energy and jumps up into a virtual excited state, it can drop down into either the vibrational excited state or the ground state, as shown below. But since the vibrational excited state has more energy than the ground state, the particle needs to vent less energy if it drops into the vibrational state. This means that it will emit a photon with less energy–i.e., a different color! This is called Stokes scattering.
Of course, an electron might start higher than the ground state–say, in the vibrational excited state. Then, when it absorbs the photon and jumps into the virtual energy state, it could drop past where it started, into an even lower energy state. In this case, the emitted photon would have more energy than the absorbed photon. This is called anti-Stokes scattering.
Both Stokes scattering and anti-Stokes scattering are examples of Raman scattering, which I explained from a classical point of view last week.
Why Two Viewpoints?
You may ask why I bothered explaining this phenomenon twice–once from a classical point of view and one from a quantum point of view. Well, both have their advantages. The classical viewpoint is undoubtedly more accessible. However, the quantum viewpoint is more accurate and, in general, more powerful in terms of making useful calculations. When we describe physical systems, scattering can quickly get insanely complicated, since the structure of any given molecule is often insanely complicated. The easiest way is to use the quantum picture of scattering and let the band structure of the material supply the allowed energy levels. When you do that, you can’t ignore momentum.
One reason I personally discussed both descriptions is that I’d like you, my readers, to see how the classical and quantum pictures correspond–how they’re the same and how they’re different. And, well…scattering is just fun!
You may find the following articles I’ve written helpful.
At some point in his or her life, almost every child on Earth asks, “Why is the sky blue?” The question is so prevalent that, to me, it has come to represent the wonder that the world holds for a a child. Adults don’t ask such questions… at least, not unless they’re scientists.
Part 1: John Tyndall
In 1859, physicist John Tyndall thought he’d found the answer to the sky’s color. His studies of infrared radiation required him to use containers of completely pure air. He discovered an ingenious way to detect whether or not his air was sufficiently pure: shine intense light through it. The light would scatter off of any dust motes or other particles, causing telltale “sparkles” that let Tyndall know he wasn’t finished purifying the air.
But Tyndall also noticed something strange. When light did scatter, it was disproportionately blue-tinted–but light that passed through the air without scattering was disproportionately red-tinted. We can see this effect in the figure below, which shows clear light passing through opalescent glass. The glass itself lights up blue, but the light that comes out the front is orange. This is called the Tyndall effect, or Tyndall scattering.
(If you want to see this effect for yourself, pour just a tiny bit of soap or milk into a glass of water and shine a small flashlight through it. The path of the light through the water will be visibly blue. There are also severalvideos on Youtube.)
In a staggering leap of logic, Tyndall extrapolated from his dust-particle experiments to the color of the sky. Perhaps, he thought, the scattering of sunlight off of particles in the air causes that familiar blue tint! By Tyndall’s time, physicists knew that white light is a wave made up of all the colors of the visible spectrum, and that one can separate the colors of light using a prism.
Tyndall’s basic idea is shown below. As (roughly) white sunlight enters the atmosphere, the blue light scatters off of dust particles in the air and spreads throughout the sky. Eventually some of it scatters down to our eyes and makes the sky appear blue. The remaining, non-scattered light is yellow or orange, and this is what we perceive as the light coming directly from the sun.
Incidentally, Tyndall’s theory also explains why sunsets are red. When the sun is parallel to the Earth, none of the Tyndall-scattered blue light reaches our eyes at all–we see only the red light left over after the rest has been scattered.
Not all the technical details of this theory are correct. It turns out that sunlight is not pure white light, but closer to a blackbody spectrum. The particles that the sunlight scatters off of are not dust particles, but rather pockets of hotter or cooler air, which act like particles due to refraction. And the fact that we see the sky as blue, rather than violet (an even shorter wavelength of light that experiences even more scattering), has more to do with how the human eye evolved than anything special about blue light itself. Nevertheless, Tyndall’s idea is essentially right–and a brilliant logical leap after a happy accident of discovery.
Part 2: Lord Rayleigh
But why does blue light scatter more than red light? And, for that matter, how does scattering work at all? If light is a wave, it can’t just bounce off of a particle, can it? (Of course, light is both a particle and a wave, but the description is still deeper than “bouncing.” We’ll talk about that in a bit.)
In 1904, John William Strutt, better known as Lord Rayleigh, examined the Tyndall effect more carefully. In the time since Tyndall, James Clerk Maxwell had discovered that light is made of electric and magnetic fields. (For a more detailed description, see my article on refraction.)
Rayleigh also knew that an atom is made up of a positively-chargednucleus surrounded by negatively-charged electrons. (As we know from Bohr, this is essentially correct.) What would happen if you were to somehow pull one of these electrons away from the nucleus? Assuming you didn’t pull too hard before you let go, the nucleus would pull the electron back in, and the electron would oscillate around the nucleus like a mass on a spring.
Of course, the electron was already (to a good approximation) orbiting in a circle around the nucleus, and it doesn’t stop orbiting after we perturb it. But because it keeps overcorrecting for the perturbation, the electron yo-yos back and forth between two elliptical orbits.
Rayleigh’s insight was that a propagating electromagnetic field–that is to say, light–pushes and pulls at the electron in the exact way necessary to make it wobble back and forth. Of course, there’s a price to pay. Wobbling the electron costs energy, which is taken out of the electromagnetic field, causing the incoming light to be absorbed by the atom and disappear.
But the electron doesn’t stop wobbling after the light is absorbed! The electron’s motion keeps tracing out the shape of the light’s electric field. And since the electron is a charged particle, this tracing-out actually recreates its electric field–producing more light of the same color as the original! To reiterate: The electron absorbs the original light, then re-emits it in a random direction. We call this behavior Rayleigh scattering.
So why is the scattered light more likely to be blue? Well, electrons interact better with light when they’re accelerating very quickly. And because blue light has a shorter wavelength than red light, it accelerates the electrons more quickly, which makes them more likely to absorb light. (I know this isn’t a terribly satisfying answer…but you’ll have to trust me when I say that it emerges in the mathematics.)
As always, I’m glossing over many, many details here. Rayleigh’s calculation only works for very small particles, which the particles in the sky are not. And it’s important that the electron doesn’t wobble at a specific frequency…otherwise, a phenomenon called resonance makes electrons absorb a much larger fraction of incoming light. (I will explain resonance sometime in the future, I promise!) And, of course, the completely correct picture is quantum mechanical in nature, but I’m going to save that for next week’s post.
Part 3: Adolf Smekal and Sir C.V. Raman
Rayleigh gave us an explanation for how light scatters off of atoms. But what about molecules? As it turns out, things don’t work the same way at all! In 1923, the unfortunately named Adolf Smekal predicted that, if light scatters off of a molecule, some of the scattered light should be a different color than the incoming (or “incident”) light. And in 1928, the brilliant experimentalist Sir Chandrasekhara Venkata Raman verified the effect. His discovery won him the Nobel prize.
In an atom, electrons are localized to one nucleus. But in a molecule, the electrons have several atoms to roam across. (As I discussed in my post on bonding, atoms in a molecule share electrons.) When an electromagnetic field comes along, it pushes the electrons into preferred positions, which causes the molecule to polarize–meaning that certain parts of the molecule are positively charged and other parts are negatively charged.
So far, this isn’t too different from electrons in single atoms. After all, our electromagnetic field moved the electron then, too. But atomic bonds in molecules are not static things. Because of the heat in the molecule, the atomic bonds wobble and vibrate all on their own. This means that once a molecule is polarized, the electrons wobble, too!
So what happens to incident light? Well, the wiggles of the electromagnetic field do indeed wiggle the electrons. But the electrons’ wiggling speed is affected by how much the molecule itself is wiggling. Thus, the wiggle that the electrons trace out to produce the new outgoing light is different than the wiggling of the incoming light alone. As a result, the scattered light can be a different color–either a higher or a lower frequency–than the original light.
(Again, there is a quantum mechanical explanation for all this, but we’ll skip it for now.)
Applications for Raman Scattering
Since the wobbling of the electrons in a molecule depends strongly on the type of atomic bonds within the molecule, Raman scattering can be used as an extremely sensitive probe of a molecule’s structure. The Raman spectrum of a molecule can even act as its identifying “fingerprint.” This is especially helpful in organic chemistry, which gives typical spectroscopy methods trouble, because organic molecules are overwhelmingly composed of the same handful of elements–carbon, hydrogen, and oxygen–but can take on incredibly complicated shapes. For example, the figures below show the Raman spectra of hexane, a relatively short string of carbon atoms with hydrogen “fingers,” and graphene, a two-dimensional honeycomb lattice of carbon atoms. (Graphene is amazing stuff, by the way…amazing enough that you should expect a whole post on it at some point.)
(A small brag: these Raman spectra plots are actual data taken by me when I was an undergraduate student. The measured graphene was even grown by me in the lab.)
If the atomic bonds change, the Raman spectrum can track that, too. (A lot of my undergraduate research involved measuring how the Raman spectrum of graphene changed when I poured acid on it.) All of this is very cool and interesting stuff…but I think I’ve written enough for now.
“Like the microscopic strands of DNA that predetermine the identity of a macroscopic species and the unique properties of its members, the modern look and feel of the cosmos was writ in the fabric of its earliest moments, and carried relentlessly through time and space. We feel it when we look up. We feel it when we look down. We feel it when we look within.”
~Niel Degrasse Tyson
A lot of news outlets are touting this as a big discovery because it is indirect evidence for gravitational waves or because it is proof of the Big Bang. But the former reason is misleading and the latter is simply wrong. It’s big news because, if true, it’s very definitive evidence for something called cosmic inflation.
There’s already a lot of news out there on the BICEP2 discovery, but I figured I’d explain my take on it, too. Hopefully I’ll be more accurate than the standard popular-science article and less technical than the standard science blogger.
Our Universe: The Early Years
(or, more accurately, The Early Millionths of a Second)
Since the inflaton is quantum mechanical, it is a wave as well as a particle. And that wave is wobbling rapidly. The quantum wobbles can be quite large in amplitude–the amplitude of a quantum wave is analogous to the height of, say, a water wave–but they only happen over very short distances. But the rapidly expanding universe stretches these wobbles out to enormous scales.
After inflation ends, the inflaton dumps all of its energy into more typical particles like electrons and protons. This process is called reheating. (“Reheating” is a bit of a misnomer. Perhaps it should be called first-heating.) And those stretched-out quantum wobbles matter. In places where the inflaton wave had a large amplitude, we got more, faster-moving normal particles. In places where the amplitude of the inflaton wave was small, we got fewer, slower-moving particles. Thermodynamically, this meant that certain parts of the early universe were much hotter than others.
Of course, the inflaton had a lot of energy, and everywhere in the universe was so absurdly hot that no atoms could form–electrons, protons, and neutrons were all torn apart by the intense temperatures. So we had lots of charged particles flying around very fast. But accelerating charged particles emit light (which happens to be how radios work). So the early universe was very, very bright. To this day, that primordial light remains. Over time it has gotten much dimmer and much redder (thanks to cosmic redshift), but still permeates the universe everywhere. We call it the cosmic microwave background,or CMB for short.
And the quantum inflaton fluctuations are still there, too. Because they affected the temperature of the early universe, these fluctuations affected the spectrum of the CMB. The light has a higher frequency where the inflaton waves had a high amplitude and a lower frequency where the inflaton fields had a low amplitude.
Although the effect is very, very small (about one part in one hundred thousand!), we can actually observe these fluctuations in the CMB, which is why the theory of cosmic inflation has become fairly mainstream among cosmologists. The figure below is a recent map of CMB by the Planck Collaboration, which shows relative temperatures of the early universe.
If you like, you can think of mass as actually causing empty space to be added or removed. Distances are shrinking or growing and angles are changing. Even though we can attribute this to the stretching or warping of space, I describe it as empty space being created or destroyed because I think it helps us understand exactly what a primordial gravity wave is. And the term itself is actually a bit misleading; the gravitational waves recently detected by the Harvard research team, are nothing like the ones which we hope to detect with laser interferometers like LIGO and LISA.
Quantum mechanics tells us that the world is inherently probabilistic, so even highly improbable things happen. Empty space is no exception: space is only empty on average. Because of quantum fluctuations, particles are constantly appearing and disappearing in so-called empty space. Indeed, space is buzzing with particles that only exist for a short period of time. These are called virtual particles. It sounds crazy, but it’s true. We even have experimental evidence.
You’ve probably heard that bit before, but here’s the clincher: Just like particles constantly appear and disappear, so does empty space. Even if there’s no mass in the universe to warp or stretch spacetime, it warps and stretches all by itself because of quantum fluctuations. This is what people mean when they talk about primordial gravity waves.
(Actually, there’s another way to think about this, related more closely to virtual particles. We already learned that every wave has a particle associated with it. Space and time can warp in a wave-like way, similar to an electromagnetic wave. And just as electromagnetic waves have the photon as an associated particle, gravitational waves have the graviton. And there are virtual gravitons that fluctuate in and out of existence, just like other virtual particles.)
And spacetime itself is affected by inflation in the same way that the inflaton is. Although the fluctuations in spacetime occur on extremely short distances, when the universe undergoes inflation, the quantum fluctuations in spacetime get expanded to enormous scales.
These fluctuations are now essentially impossible to see directly, but we can look for their signature in the CMB. As the CMB photons pass through areas where spacetime is warped, they change polarization depending on how extreme the warping is. I won’t go into detail about what polarization is right now (in a later article, I promise!), but suffice to say that it is a property of light and we can represent it as an arrow perpendicular to the direction a photon is traveling, as shown below. And when we talk about the direction of polarization, we are talking about the direction this arrow is pointing.
As a further test of inflation (and a probe into what caused inflation), we can try to observe the polarization of the light in the CMB. At each point in the sky, we would measure the polarization and describe it as an arrow pointing in the direction of polarization. Unfortunately, this measurement is fiendishly difficult… much more difficult than measuring the frequency and intensity of the light from the CMB.
However, this is precisely what the BICEP2 team claims to have achieved. They couldn’t measure the entire sky like Planck did for the temperature associated with the CMB, but they did measure a small piece of the sky (plotted below). The lines represent the directions of polarization. The colors represent the polarization’s “B-mode pseudoscalar,” which measures how much the lines form a spiral shape. The pseudoscalar patterns that BICEP2 observed is characteristic of primordial gravity waves.
The BICEP2 results are extremely strong evidence that our understanding of the universe after the Big Bang is correct and that cosmic inflation did indeed happen. Up until this point, inflation has been somewhat controversial. It successfully makes predictions, but it has some conceptual problems. Observation of primordial gravitational waves would put this controversy to rest. These observations can also offer insight into how inflation started. Understanding how the inflaton grabbed the tiny fluctuations in spacetime and expanded them will help us understand the inflaton a lot better.
Finally, the BICEP2 results are the first real measurement we’ve ever made of quantum gravity. Describing the quantum fluctuations in spacetime is tricky business and we really don’t have a good method for it. This is a huge issue in physics at the moment, called the “problem of quantum gravity.” However, in some special cases, where space and time are relatively well behaved (in a technical sense) and where the fluctuations are small, we can come up with a good mathematical description. This kind of math leads to some pretty mind-boggling things, such as Hawking radiation. But if we try to go beyond the simplest cases, the math blows up in our faces. A measurement of primordial gravitational waves tells us that, at least in the simplest cases, we’re on the right track.
By assuming a cause for inflation, cosmologists have been able to analyze temperature measurements made in the past (like the Planck map above) and propose a rough upper bound on how much of a signal we should see from primordial gravity waves. It looks like the BICEP2 results violate this upper bound. This isn’t necessarily a bad thing–indeed, it makes inflation more certain, not less, and perhaps implies new physics. But it does mean that the scientific community is fairly skeptical of the BICEP2 results.
Fortunately, BICEP2 isn’t the only telescope on the job. A huge number of other collaborations are trying to study the polarization of the CMB. (My good friend Sara Simon is part of the Atacama B-Mode Search team, for example.) BICEP2 is just the first group to gather and analyze their data. Once the other teams finish gathering and analyzing their data, we’ll be able to say for sure whether or not BICEP2′s conclusions were correct.
There is a lot of information related to the BICEP2 results out there. If you’re curious, here’s some more to read about them.
The connection between symmetries and conservation laws
is one of the great discoveries of twentieth century physics .
But I think very few non-experts will have heard either of it or its maker[:]
Emily Noether, a great German mathematician.
But it is as essential to twentieth century physics
as famous ideas like the impossibility of exceeding the speed of light.
It is not difficult to teach Noether’s theorem, as it is called;
there is a beautiful and intuitive idea behind it.
I’ve explained it every time I’ve taught introductory physics.
But no textbook at this level mentions it.
And without it one does not really understand why the world is such that
riding a bicycle is safe.
At first, I had planned to talk about Rayleigh and Raman scattering today. However, in honor of International Women’s Day, I changed my mind. I wanted to write about the woman that Albert Einstein called the greatest female mathematician of all time, Emmy Noether.
One of the Greatest Mathematicians of All Time
Noether’s accomplishments are as incredible as they are varied. She made many seminal contributions to the field of abstract algebra and one incredible contribution to the field of physics. Her first published work helped solve the “finite basis problem,” a major open problem in mathematics at the time–even if she later called the work “crap.” (She had high standards, to say the least.) And in her later work, she:
In short, whenever Emmy Noether approached a mathematical problem, she invented a whole new field of study. Almost no one in history has been so successful. As mathematician Nathan Jacobson said,
The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries.
And she was as brave and audacious as she was brilliant. When Noether first attended the University of Erlangen, women were forbidden from taking courses. Instead, Noether had to ask each individual professor for permission to audit his class. She was one of only two women who attempted to study there in this way.
Later, with the help of mathematical giants David Hilbert and Felix Klein, Noether became the first woman lecturer (and later professor) at the University of Gottingen, much to the distaste of several other faculty members. But at first, Noether worked for no pay and her lectures were advertised under Hilbert’s name; officially, she was his “assistant.”
I can’t possibly describe everything that Emmy Noether accomplished. So instead, I will devote the remainder of this post to describing the Noetherian idea I understand best, one of the most important ideas in modern theoretical physics: Noether’s theorem.
If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.
Right now, this statement doesn’t make any sense. We need to dissect it and understand what “a continuous symmetry property” is and what it means to be “conserved in time.” We’ll discuss each of these ideas in turn. Let’s start with symmetry.
Symmetry: The Science of Sameness
To learn about symmetry, we have to go all the way back to elementary school. Bear with me; the end result may be simple, but it is completely unintuitive. First, picture a simple square in your head. Now imagine rotating it by forty-five degrees counter-clockwise, as shown below. The square looks different now. Suddenly it’s a diamond.
This is pretty unsurprising. We’re all familiar with this. (Indeed, this is an example where symmetry is not preserved.) But now let’s imagine the same square and rotate it by ninety degrees instead of forty-five. The behavior is qualitatively different. The square looks exactly the same… as if we hadn’t rotated it at all.
Incidentally this would remain true even if we rotated the square by degrees or by degrees or by degrees. Rotating by ninety degrees always returns the square to itself. Thus we say that a rotation by ninety degrees is a “symmetry” of the square.
Much in the same way that we quantify the amount of things by the counting numbers (e.g., I have two pencils or four apples), mathematicians quantify the amount of symmetry something has using “symmetry groups.” The symmetry group of an object is the collection of operations under which that object remains the same. For example, the symmetry group of the square contains every rotation that is a multiple of ninety degrees.
A symmetry group has many special properties, ones that probably seem familiar and intuitive from the world of numbers. In particular, they behave very similarly to the addition of integers we’re all familiar with–except that they may not commute. . However, if I rotate a cube by ninety degrees clockwise and then ninety degrees towards me, the composite operation is different from the operation where I rotated by ninety degrees towards me first and then ninety degrees counterclockwise.
But I digress. Let’s return to just two dimensions. Instead of a square, let’s imagine a pentagon, which is symmetric under rotation by degrees, degrees, degrees, degrees, and degrees.
Let’s add another side. A hexagon is symmetric under rotation by degrees, degrees, degrees, degrees, degrees, and degrees.
Two things are happening each time we add a side: The smallest rotation operation under which the shape is symmetric shrinks–each time we add a side, the angle is smaller–and the number of rotation operations under which the shape is symmetric grows. Indeed, the number of rotation operations under which the shape is symmetric is equal to the number of sides.
What happens if we add infinite sides? A polygon with infinite sides is a circle. In this case, our shape is symmetric under a rotation by any angle at all! This is what we call a continuous symmetry, and this is what Noether means in her theorem when she says a system has a continuous symmetry property.
Just to drive the point home, let’s look at some examples in three dimensions. Like the circle, a parabola is symmetric under any rotation around its central axis. A sphere is even more symmetric–it is preserved by any rotation at all around its center. But something doesn’t have to be round to be preserved by a symmetry. An infinitely large, flat sheet of paper is symmetric too. The operation that preserves it is motion in a straight line. If I am an ant on this infinitely large sheet of paper, and I walked twenty meters in any direction. I might never now exactly how far I walked…because everything looks the same. That’s a symmetry, too!
Sameness in Time
Now we know what a continuous symmetry property is. What does Noether’s theorem say this implies? The easiest way to understand this, I think, is by example. Let’s go back to the parabola, which we know is symmetric under rotation about its axis. Now let’s imagine dropping a marble into the parabola. If we drop the marble straight down it will oscillate back and fourth across the parabola under the force of gravity, just like Bart Simpson in a half-pipe, as shown below.
But if we throw in the marble in with just a little bit of spin, so that it enters the paraboloid traveling a little bit north or south (as opposed to up or down or radially outward or inward), we get totally different behavior where the marble appears to clamshell around the paraboloid:
(Note that although the qualitative behavior of these animations is correct, it isn’t the exact solution. It wouldn’t be too difficult to solve the equations of motion–it’s an undergraduate classical mechanics problem–but I didn’t do so. Call it an exercise left to the reader. )
Now the marble’s motion is far from constant. It speeds and slows, bobs up and down. However, it turns out that there is a property of the ball’s motion that never changes. The exact product of the mass of the marble, times its distance from the origin, times how fast it rotates about the origin, will always stay constant. This product is called the angular momentum of the ball, and it never changes as the ball wobbles and bobs and accelerates and decelerates. Because it is constant in time, we say that the angular momentum of the marble is conserved.
Noether’s theorem tells us that the reason that the angular momentum of the ball is constant in time is because the paraboloid is symmetric under rotation. Every symmetry of the space our marble lives in generates a conserved quantity of the motion of our marble–something about it that is constant in time. Of course, these conserved quantities are rarely obvious. They’re usually some product of things like mass, acceleration, and even position.
But Why? The Parable of the Rockies
Now that we know what Noether’s theorem says, can we get some intuition as to why it should be? My hometown of Boulder, Colorado is nestled in the eastern foothills of a huge mountain range, the Rocky Mountains, which span the entire western horizon from north to south. To the east of Boulder, Colorado is completely flat.
When settlers first traveled west towards Boulder and saw the mountains in the distance, they rejoiced. They believed that by nightfall they would be safely nestled in the foothills. But it was not to be. When night fell, the world looked exactly the same. The settlers appeared to have made no progress towards the mountains. The settlers were discouraged, but they carried on. They were sure they would be in the foothills by nightfall the next day. But again, it was not to be. The next morning, the mountains were as eternally distant as ever. This went on for two weeks before the settlers reached the foothills.
What happened was that the mountains were so vast, one could see them from a great distance away. To a good approximation (from the settlers’ point of view), they were infinitely wide and infinitely tall. Despite this, though, the settlers still might not have been fooled if eastern Colorado weren’t so absurdly flat. With no landmarks to reference, the settlers were tricked into believing that nothing had changed as they traveled–that perhaps they hadn’t traveled at all. The only landmarks were the unreachable mountains.
Because the settlers were essentially on an infinite flat plane, the world was symmetric under motion along a straight line. As they traveled along that line, the world continued to look the same to them, so they were unable to tell that time had passed. A particle moving on an infinite plane will behave exactly as it did before it moved, since nothing in its environment has changed–thus, momentum is conserved. The same goes for a particle moving in a circle of constant radius along the surface of a paraboloid, along one of the great circles along the surface of a sphere, or in the direction of symmetry along the surface of any other corner-less, three-dimensional shape. (More precisely, the surface must be “smooth,” which is a term defined as “infinitely differentiable.”)
This isn’t so surprising. But what is surprising is that, no matter what the particle does, the component of its motion along the direction of symmetry (e.g., around the axis for a paraboloid or around the center for a sphere) remains unchanged. And this corresponds to a property of the particle’s motion that is unchanged in time.
Just to recapitulate that last bit: A symmetry in space corresponds to a symmetry in time. If nothing changes as I travel in space, then I can’t tell that time has passed. So some aspect of me ceases to change in time.
At the end of the day, Noether’s theorem is beautifully, surprisingly simple…and deeply profound.
At first glance, Noether’s theorem just seems like an esoteric quirk of geometry. But its implications are very deep and very far reaching. Let’s step back a little and look a few centuries into the past. Newton’s first law of motion is that:
An object at rest tends to stay at rest and an object in motion tends to stay in motion, unless acted on by an external force.
Well, that’s a statement about a conserved quantity! Newton is telling us that there is some property of the motion, in this case momentum, that doesn’t change in time. Newton didn’t know about Noether’s theorem. But it turns out that there exists a symmetry that generates Newton’s first law: translation invariance. Empty three-dimensional space looks the same everywhere you go. If you were a bird–or a particle–and could fly in any direction in a straight line, everything would appear the same no matter how far you went. This symmetry is what generates the conservation of momentum described in Newton’s first law! (As a side note, Noether’s theorem also only holds in the absence of external forces, so Newton’s law is consistent with that.)
And replacing Newton is just the start! Since its discovery, Noether’s theorem has become an integral part of theoretical physics. In particle physics, the symmetries of a system generate particles, and Noether’s theorem has inspired the discovery of many subatomic particles. In general relativity, Einstein’s equations are so difficult that an exact solution is often unsolvable without the aid of symmetry–but Noether’s theorem allows us to find the spacetime symmetry related to a quantity we believe to be conserved.
It’s not an understatement to say that Noether’s theorem is one of the most important developments in theoretical physics in the last two hundred years. And this theorem is only one of Noether’s myriad brilliant achievements. Yet I’d be willing to bet that most of you never heard her name in school.
The New York Times did an article on Emmy Noether here.
Unfortunately I will be taking a hiatus on blog posts until mid December. The reason is that graduate school is pretty hectic at the moment and I’m feeling a bit too overwhelmed. See you all in about three weeks!
I have one simple request.
And that is to have sharks
with frikkin laser beams
attached to their heads!
Always look on the bright side
…unless you’re holding a laser pointing device.
The laser is, without a doubt, one of the most ubiquitous, archetypal technologies of modern times. And it is one of the most direct applications of quantum mechanics. But how do lasers work?
It All Starts In The Atom
The story starts deep within the atom. I’ve previously discuss the fact that particles are waves and that this forces electrons to have only certain specific energies inside an atom. The energy and momentum of a particle control how many times the corresponding wave wiggles. And these must fit in a circle around the nucleus of the atom, as shown below.
If the atom is part of a molecule, especially a crystal, the discrete allowed energies become so numerous that they look like continuous bands. And this leads to band structure.
For clarity, physicists often imagine extremely simple atoms with only two or three allowed electron orbits, each of which is allowed only at a single specific energy and a single specific momentum. We then plot these energies as a function of their allowed momenta. The plot is called an “energy level diagram,” and it looks something like the figure below.
Between Light And Matter
Now let’s imagine an electron sits in the lowest energy level, as shown below.
When a photon—a light particle—hits the atom (or alternatively passes right through it), it has the potential to affect the electron. Classically (i.e., without quantum mechanics), the light would accelerate the electron, since the electron is a charged particle and light is made up of electromagnetic fields and electromagnetic fields affect charged particles. However, if the electron accelerated, it will gain kinetic energy. This gain is only allowed if the electron ends up with one of the allowed energies.
If the electron is accelerated, it will absorb the photon, absorbing both the energy and momentum of the photon. So it is only allowed to absorb the photon if the electron’s new energy and momentum are allowed within the atom. Otherwise, surprisingly, the photon passes right through the atom unmolested, as shown below.
The same process works in reverse. Electrons are lazy and they want to be in the lowest possible energy state. So they’ll do whatever they can to drop from a high energy state to a lower one. And the easiest way for an electron to drop to a lower energy state is by
emitting a photon. The emitted photon must, of course, have energy and momentum such that the electron’s new energy state is allowed, as shown below. This process is known as fluorescence.
The rules determining how an electron may change energy and momentum are called “selection rules.”
Cheating Selection Rules
Of course, selection rules aren’t absolute. Quantum mechanics is inherently probabilistic, and the Heisenberg uncertainty principle forbids us from knowing all quantities perfectly well. This means that if we shine a beam of light on an atom such that most of the photons have the wrong energy and momentum for the electron to transition to a new energy level… every once and a while, by pure quantum chance, a photon will come along with the right energy and momentum and the electron will transition, as shown below.
Another way you can think about it is that, eventually, the electron itself moves a little bit out of the allowed energy levels and it can absorb one of the forbidden photons, as shown below.
Now, let’s imagine that an electron starts in a low-energy state. And it is excited into a high energy state by a photon with the appropriate energy and momentum. Then, while the electron is still in this high-energy state, another photon with the same energy and momentum hits the atom. What happens?
Intuitively, the photon should pass harmlessly through the atom, unabsorbed, because the electron has nowhere to go. However, this isn’t what happens at all. The electron will drop down to a low-energy state and emit an identical photon, traveling in the same direction and with the same energy and momentum as the incident photon, as shown below. This is called stimulated emission, and it is the magic that makes lasers work.
Unfortunately, I can’t really give a good explanation for how stimulated emission works. The mathematics behind it, and that predicts it comes from time-dependent perturbation theory, a way to examine the quantum mechanics of complicated situations. I can say that absorption and stimulated emission are opposites. The math for each is the same. Indeed, process that’s most different is the most intuitive: fluorescence, where the atom decays without any stimulus at all.
Population Inversion and Gain
If we could take advantage of stimulated emission, we could use it to amplify a beam of light and make it very intense. More importantly, ever photon in the beam could be generated from a single seed photon. The beam could be made of clones, all traveling in the same direction, all with the same energy and momentum. This would let us control the properties of the beam very precisely. (This property is called coherence.)
Unfortunately, atoms like to fluoresce, which means that most electrons do not stay in a high-energy state for long enough for us to initiate stimulated emission. Is there a way around this?
There is a way around this problem! Some transitions between states take longer than others. (This has to do with the quantum mechanics of selection rules that I talked about earlier.) Furthermore, some transitions are more likely to occur naturally than others. In other words, if we select the right atom, we can control how electrons in it transition between states. We can find an atom where the electrons transition to a high energy state very quickly, but then decay into a middle state where they stay for a long time. If we do this fast enough, we can get all of our electrons into the middle state, as shown below. This is called a “population inversion.”
Once we have a population inversion, all it takes is one seed photon. We put a block of our inverted material (called a gain medium) in between two mirrors, as shown below. Then we make the material fluoresce once. It doesn’t really matter how. Eventually the material will fluoresce if it’s in population inversion.
Once one photon is between the two mirrors, it will bounce off of a mirror and pass through the gain medium, causing stimulated emission. Then two photons will bounce off of a mirror and pass through the gain medium, causing stimulated emission. Then four photons will bounce off of a mirror… Well you get the idea.
This is how laser light gets so intense.
But why is laser light only one color? This is actually much easier to explain. It’s a consequence of the fact that the gain medium is placed between two mirrors. Remember that photons are both particles and waves. And that the wavelength of the wave determines the color of the light. Moreover, light waves are made up of electric and magnetic fields. The electric field of the light must be zero at the mirror, because mirrors are conductors. The electrons in the mirror move to cancel whatever electric field might otherwise exist.
This means that, just as an electron orbiting a nucleus can only fit an integer number of wavelengths into the orbit, a light beam can only fit an integer number of wavelengths between the two mirrors, as shown below.Otherwise, the wavelength would not be zero.
Where don’t lasers have applications? We use them in medicine for laser eye surgery. We use them in our computers to read optical disks. We use them in our factories to cut metal. We use them to send light signals through fiber optic cables for communication. We use them to measure distance. We use them to measure time. We use them to generate fusion power, and we use them to help us calibrate our telescopes. I’ll talk about some of these ideas in future posts. If you’d like to hear about a specific application, let me know and I’ll see what I can do.
Where to even start? Here are some resources:
PHET has a simulation of a laser suitable for classroom demonstrations. It just runs in a web applet.
Minute Physics has a nice video. It uses Bosonic statistics to explain stimulated emission. I don’t really like this explanation, but it does give a good intuition.
The patient accretion of knowledge,
the focusing of all one’s energies on some problem in history or science,
the dogged pursuit of excellence of whatever kind
these are right and proper ideals for life.
Nothing can escape from a black hole, not even light. This is why we call them “black.” One would imagine, then, that black holes are black invisible menaces, lurking out in the depths of space. Surprisingly, though, black holes glow. The cover image shows a radio photograph of the center of the Milky Way. The center glow, Sagittarius A, is partly due to a supermassive black hole, Sagittarius A*. (No, that doesn’t lead to a footnote…the name of the black hole actually is Sagittarius A*, pronounced “a star.”)
Black holes glow because they are very messy eaters. As a black hole sucks in surrounding matter, it pulls its food into a disk or a sphere around it, called an “accretion disk” or an “accretion shell,” as shown below. And it is partly this disk that generates the incredible glow. (There is another process, called a “jet,” which also produces a lot of light. I’ll briefly talk about it later.)
But why doesn’t stuff in the accretion disk just fall into the black hole? The answer, elegantly enough, is the same reason that the planets in our solar system don’t fall into the sun.
Imagine that you tie a ball to a string and spin it over your head. The ball will fly out to stretch the string as much as possible and, if you let the string go, the ball will fly away from you in a direction tangential to the circle. This effect is so prominent that it can be used to make a weapon called a “bola.”
As Sir Isaac Newton predicted, objects like to travel in straight lines–you have to push or pull them to make them deviate. This resistance to change is called momentum. Thus, to make an object travel in a circle, you have to constantly pull it towards the center of the circle, forcing it to turn. The faster an object moves (or the more massive it is), the harder it is to turn, and the more force you have to use to pull it towards the center of the circle. Although the object’s tendency to fly out of the circle emerges purely from its momentum, for convenience, we often pretend it’s a separate “centrifugal force.”
Matter in accretion disks is often spinning too fast to fall into the black hole. The gravitational pull of the black hole isn’t strong enough to counteract the centrifugal force of the matter–partly because the black hole is spinning too and drags the matter with it, partly because the matter was spinning to begin with. (On the cosmic scale, most things in the universe are spinning.)
Over time, the black hole does win. The matter does lose outward momentum and fall into the black hole. (Like energy, momentum can’t be created or destroyed, but it can be transferred. Most of it is vented through the “jet” light-creating process that I’ll briefly explain later.) However, as stuff falls into the black hole, the gravitational pull of the black hole accelerates it up to incredible speeds, which in turn heats it up to incredible temperatures. And hot matter glows.
(Temperature actually contributes to the glow in another, less direct way. The in-falling matter is often so hot that it ionizes, its electrons separating from their nuclei. These charged particles follow the spin of the disk they’re in, which causes them to accelerate. Since accelerating charges emit light–which, incidentally, is how radios work–the disk glows even brighter.)
The glow has another surprising effect, though. We often imagine accretion disks to be very thin, flattened out by the spinning of the disk and the black hole, the same way that a pizza chef flattens out dough by spinning it. But they’re actually a bit thick. The secret is light.
A Quantum of “Push”
In the time of Sir Isaac Newton, there were two competing ways of understanding light. Newton believed that light was made out of tiny particles called “corpuscles” that carried kinetic energy and momentum and bounced off of things like any normal particle. In contrast, Christiaan Huygens believed that light was like sound: a wave that propagated through a clear medium, like air or glass.
Imagine that a beam of light bounces off of a mirror, as shown below. One way to describe this is by using the equations of optics and electromagnetism. However, another way is to imagine a bunch of physical particles–which we now call photons–hitting the mirror and bouncing off of it. But Newton tells us that “for every action there is an equal and opposite reaction.” When the mirror pushes the photons, the photons must push back.
This effect is called radiation pressure. We don’t usually notice it because each individual photon doesn’t carry much energy compared to a human being. We need a lot of them to exert an appreciable force. However, we can harness radiation pressure to do some pretty cool things. The solar sail proposal for space travel is based on this idea.
(Experts know that the conception of light as a wave also predicts that it carries energy and momentum. However, we need to treat light as an electromagnetic wave, governed by Maxwell’s equations. Particle-wave duality lets me explain radiation pressure a lot more easily.)
Why Accretion Disks are Thick
So what does radiation pressure have to do accretion disks? As we now know, the matter in the accretion disk is producing quite a lot of light. When this light scatters, it exerts an outward force on the in-falling stuff, partly counteracting the pull of gravity and the flattening effect of the spin. If enough photons hit the in-falling gas, something amazing happens: the matter stops falling. The constant radiation pressure from within the disk completely counteracts the force of gravity.
The point when the glow of the accreting matter is bright enough to stop it from falling into the black hole is called the Eddington limit, after Sir Arthur Stanley Eddington. With rare exception, we never see accretion disks glowing brighter than this; if there’s enough glow to cause that, it means more matter is flying outwards than inwards, so the disk dissipates and the glow subsides. (The Eddington limit is usually lower than the brightness required to completely counteract gravity. The radiation pressure has some help from the centrifugal force, as discussed above.)
This is also why accretion disk are thick. The force of gravity and the incredible spin of the black hole should flatten the disk out like a pizza crust, and to a good extent, it does. However, the light from the glow of the disk pushes the matter outward and puffs it up a little bit, so that it looks more like a slightly squished donut. (Accretion disks seem to fall into several categories of shape–some thicker, some thinner. The factors involved are an ongoing area of research, but radiation pressure is often important.)
In the case of rotating black holes, there’s another source of light, the so-called “jets.” The plasma physics of the disk accelerates the in-falling matter to enormous velocities, ultimately launching it into space around the poles of the black hole and along the axis of rotation. These incredibly powerful jets of matter, which glow for the same basic reasons of centrifugal force as accretion disks, are another reason black holes are easy to spot. They also allow matter in the accretion disk to bleed off its outward momentum enough to fall into the black hole.
What I’ve given you is a very simplistic introduction to a very rich and difficult topic. Accretion physics is still an active area of research. To truly understand what’s going on, we need to simulate what happens to the stuff in the accretion disk, taking fluid dynamics, electromagnetism, and general relativity into account. I’ve tried to find some non-technical resources.
I am by no means an expert on accretion physics, so I could have gotten something wrong here. If I have, please bring it to my attention! And if you have any questions, please bring those to my attention, too–I’ll do my best to answer them!