Probability: Part 2 (Distributions)

A nice breakfast with the bell curve.
A nice breakfast with the bell curve. (source)

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!

The Problem

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.

Galton's Bean Machine
The Bean Machine is a simple machine which is characterized by the Binomial Distribution.

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.

Possible Path through Bean Machine
A possible route through the bean machine.

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.

Multipath in bean machine
Two possible paths leading to the same bin within the bean machine.

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 0.5^n, which turns out to be 1.56\% 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 n pegs, the ball may go right once at any of those points; this gives us n different paths. This means the total probability is n (0.5^n) which in our example is 9.38\%. The third bin from the left is a little more tricky. It turns out it requires two right bounces. This means there are \binom{n}{2} different paths. Here, \binom{n}{k} is the binomial coefficient or choose function. It gives the total number of ways you can choose k objects from a total of n. The algebraic form of the choose function is

    \[\binom{n}{k} = \frac{n!}{k! (n-k)!} .\]

This allows us to know how many total paths there could be. In the case above, n = 6 so the total number of paths is 15. The probability is therefore \binom{n}{2} 0.5^6 which is 23.4\% in our example. This trend continues in the same fashion which gives us a general form for bin k to be

    \[P(n,k) = \binom{n}{k} 0.5^n.\]

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:

    \[P(n,k) = \binom{n}{k} P_L^k P_R^{n-k},\]

where P_L is the probability of the ball falling to the left and P_R is the probability of the ball falling to the right. This is known as the binomial distribution.

Probability Distributions

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:

An example Normal Distribution
A normal distribution with a mean of 70 and standard deviation of 10.

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.

Increasing Binomial Distribution
This animation captures the way a binomial distribution with increasing n will begin to look like a normal distribution.

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 \mu. In addition, the width of the normal distribution is also characterized. This term is called the variance or standard deviation, \sigma. 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:

Normal Distribution STD Chunks
The Normal Distribution Cut into standard deviation chunks.

and we can express the functional form as

    \[f(x, \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{\sigma^2}}.\]

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.

Portion of Normal Distribution
A normal distribution for test with 60% mean and 10% stdev. 15.9% of the test takers are represented as being between 70% and 100%.

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:

Particle In Box (Lowest Level)
The lowest level distribution for a particle in a box.

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:

Particle In A Box (50% Chance of Finding The Particle)
50% of the time, a particle in a box in it’s lowest energy level will be found within the shaded area.

The second energy level is a bit more curious of a distribution, it looks like this:

Particle In Box n=2 Distribution
The second energy level for a particle in a box.

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.

Further Reading

Jonah says: If you liked Mike’s post, you might also enjoy the articles I wrote about quantum mechanics.

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Post Delayed

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.

Savannah Georgia
Savannah Georgia

 

To tide you over, I’ll put up a guest post by my good friend Michael Schmidt soon. If you follow the blog, you know him. He wrote the post on probability.

See you all soon!

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Scattering Part Two: A Quantum of Scattering

We come spinning out of nothing,
scattering stars like dust!

~Jalāl ad-Dīn Muhammad Rūmī

raman spectrometer
The laser end of a Raman spectrometer, a device designed to measure the Raman scattering of a material. (Source: The Martin Suhm Research Group)

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.

Before we can talk about scattering, though, we need to review some important ideas from quantum mechanics: energy levels and the Heisenberg uncertainty principle.

Energy Levels

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!

An electron around an atom
If an electron’s wavelength is too short (left) or too long (right), then it doesn’t fit at a given radius of orbit. However if the wavelength is an integer value of some special number (center), the electron fits.

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.

Ground state electron. Chillin.
A basic two energy level atom with one electron (yellow) sitting in the lowest energy state.

When a photona 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.

To absorb or not to absorb? That is the question.
Left: A photon with the right energy and momentum (green) hits the atom, causing the electron (yellow) to absorb the photon and jump to a new allowed energy band.
Right: A photon with the wrong energy and momentum (blue) hits the atom. The electron (yellow) is unable to absorb the photon because the electron’s new energy and momentum would not be allowed, so the photon passes through unmolested.

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.)

 Uncertainty

There’s a lot to say about Werner Heisenberg’s famous uncertainty principle…and I have said a fair amount in the past. For now, though, we’ll be brief. The uncertainty principle is a consequence of the fact that matter is both a particle and a wave. If you’ve heard of it, you probably know what it says:

    \[\Delta x \Delta p \geq \frac{\hbar}{2}\]

or, in English,

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

    \[\Delta E \Delta t \geq \frac{\hbar}{2}\]

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.

This figure is VIRTUALLY perfect!
An electron actually CAN absorb a photon of the wrong energy, if only briefly. It jumps into a virtual excited state and then drops back down to its original state, emitting a photon of the same color as the absorbed one, but in a random direction.

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.

Scattering in the classical and quantum pictures
Left: Rayleigh scattering in the classical picture. Right: Rayleigh scattering in the quantum picture. This won’t make sense unless you read my previous post.

Enter Raman

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.

The energy of a wobbling molecule
An energy diagram for an example wobbling molecule. Now we have kinetic energy states, which come from the wobbling of the atomic bonds in the molecule, in addition to the atomic energy 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.

stokes_scattering
Stokes scattering. First (left) an electron (yellow) absorbs a photon (blue) and jumps into a virtual excited state. Then (right), the electron drops down to the vibrational excited state and emits a photon of lower energy (green).

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.

Anti-Stokes Scattering
Anti-Stokes scattering. First (left), an electron starts in the excited vibrational state, absorbs a photon, and jumps into a virtual excited state. Then (right), the electron drops down into the ground state, emitting a photon with higher energy than the absorbed photon.

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! :)

Further Reading

You may find the following articles I’ve written helpful.

Other Resources

Raman scattering is a pretty esoteric topic, so all the resources this time are pretty technical. The best I could find are some lecture slides from various universities:

Questions? Comments? Insults?

As always, if you have any questions, comments or corrections–or if you just want to say hi–please drop me a line.

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Why The Sky is Blue: Lord Rayleigh, Sir Raman, and Scattering

The Sky is the Daily Bread of the Eyes
~Ralph Waldo Emerson

An advertisement for optical scattering
Why is the sky blue? Why is a sunset red? How does light bounce off of a molecule? Does it always work the same way? The great Lord Rayleigh (top left) and Sir Chandrasekhara Venkata Raman (bottom left) answered these questions. (Photographs from Wikipedia.)

 

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

John Tyndall
A sketch of John Tyndall, eminent experimental physicist in the 1800s. Note the epic beard. (Image courtesy of Wikipedia.)

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.

Tyndal effect in opalescent glass
Tyndall scattering in opalescent glass. The scattered light is blue, but the transmitted light is orange. (Image from Wikipedia.)

(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 several videos 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.

dispersion prism
White light is made up of all the colors of visible light and can be re-separated into its component colors by a prism. (Image from Wikipedia.)

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.

Grace Hopper observes the TYndall effect.
Rear Admiral Grace Hopper observes the Tyndall Effect in the sky. Light emerges from the sun (1) in every color of the rainbow (2). The light scatters off of particles in the air (3) and the blue light bounces off in all different directions, while the red and yellow light safely travels straight to the surface (4b). The scattered blue light eventually reaches the surface after scattering off of many, many particles (4a) and reaches the surface, where Hopper sees a yellow sun and blue sky (5). (Grace Hopper Image from the Anita Borg Institute for Women and Technology.)

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.)

James Clerk Maxwell
James Clerk Maxwell. Again, note the epic beard (source).

Maxwell discovered that these fields can feed into each other and become self-sustaining. A changing electric field produces a magnetic field, which produces an electric field when it changes, and so on. Moreover, when these fields oscillate like this, they behave exactly like light–meaning that light is a wave made up from these fields! (I am, of course, glossing over the fact that light is both a particle and a wave. From the quantum perspective, the electromagnetic wave describes the probability of detecting a photon at a given place and time.)

Light as an electromagnetic wave
Light as an electromagnetic wave. The red lines represent an electric field and the blue lines represent a magnetic field. A changing electric field induces a changing magnetic field which, in turn, induces a changing electric field. (source).

Rayleigh also knew that an atom is made up of a positively-charged nucleus 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.

Mass on a spring
An electron attracted to an atomic nucleus behaves much like a mass on a spring (source).

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.

atom oscillation
If an electron orbiting around a nucleus is gently pulled away from the nucleus, its original orbit will be perturbed and the electron will oscillate around the nucleus between two new 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.

Adolf Smekal
Adolf Smekal, predictor of Raman scattering, but usually forgotten.  (Image courtesy of the American Institute of Physics.)

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.

A typical polarized molecule
A typical polarized atom. Pink represents an absence of electrons and thus a positive charge, while blue represents more electrons and thus a negative charge. Black represents atomic bonds.

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!

A wobbling molecule
When a molecule is polarized, the wobbling of the atomic bonds also drives the motion of the electrons.

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.)

The Raman spectrum of hexane.
The Raman spectrum of hexane. The horizontal axis shows the difference between the scattered light’s frequency and the incident light’s frequency. The vertical axis shows the intensity of the scattered light. (Molecule image from Wikipedia.)

(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.)

Raman graphene
The Raman spectrum of graphene. The horizontal axis shows the difference between the scattered light’s frequency and the incident light’s frequency. The vertical axis shows the intensity of the scattered light. (Molecule image from Wikipedia.)

 

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. :)

Further Reading

Questions? Comments? Insults?

As always, if you have any questions, comments or corrections–or if you just want to say hi–please drop me a line.

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BICEP2, Primordial Gravity Waves, and Cosmic Inflation

“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

BICEP2 sunset
The BICEP2 telescope on the South Pole. This is the device which may have finally discovered primordial gravitational waves. (Credit: the BICEP2 collaboration)

There was some very big news today! If you haven’t already heard, the BICEP2 research group at Harvard has found evidence of ancient gravitational waves in the sky.

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)

Most cosmologists believe that in the very early universe, about 133.6 billion years ago, the universe underwent a period of extremely rapid expansion called cosmic inflation. (Although the opinion is still controversial, there are good reasons to believe inflation occurred.) In the standard story, inflation is caused by a particle called the inflaton, which–much like today’s dark energy–has some strange properties like negative internal pressure that are only possible due to quantum mechanics.

The history of the universe
A modern picture of the history of the universe, including inflation. That dramatic widening of the universe right after the Big Bang is inflation. (Image credit goes to Rhys Taylor of Cardiff University, via the Planck collaboration.)

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.

Planck Survey of the Sky
Planck survey of the CMB. The oval is the observable sky. Orange areas are hotter and blue areas are colder. The scale is exaggerated to make the contrast between high and low temperatures more obvious. However, the difference is actually only about one part in one hundred thousand. Image from the Planck Collaboration.

Wibbley Wobbley Spacetime

As I’ve discussed before (at great length, since it’s one of my favorite topics), gravity is caused by the warpings and wigglings of space and time. We think of space as stretching, shrinking, and warping based on the mass and energy in the universe. This means that the shortest possible path between two points may not be what it appears. And so, even though particles all travel along the shortest possible paths through space and time, the paths can appear curved to our simplistic three-dimensional Euclidean eyes.

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 Wobbles

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.

This is how polarization works!
We can represent polarization as an arrow (blue) perpendicular to the direction a photon is moving (red).

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.

BICEP2 polarization map
The polarization of the cosmic microwave background due to primordial gravity waves, as measured by the BICEP2 team. The lines represent the directions of polarization and the colors represent the B-mode pseudoscalar of the polarizations.

Implications

First and foremost, the BICEP2 results are not the first indirect measurement of gravitational waves. The first indirect measurement of gravitational waves won the Nobel prize. They are also not evidence for the Big Bang theory. The CMB and the expanding universe are evidence enough for that.

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.

Tensions?

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.

Further Reading

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.

 Related Articles

If you’re confused, here are some articles I’ve written in the past:

Questions? Comments? Insults?

Although my research is in gravity, I don’t do active research in cosmology. So if you know better, please correct me! (And, as always, please ask any questions you may have.)

Posted in Condensed Matter, cosmology, Physics, Quantum Mechanics, Relativity, Science And Math | Tagged , , , , , , , , , , , , , , , , | Leave a comment

International Women’s Day Spotlight: Emmy Noether

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.

~Lee Smolin

Emmy Noether Promotional Poster
Emmy Noether watches over the physical ramifications of her ground-breaking ideas. (From left to right: Emmy Noether, courtesy of Wikipedia; particle paths in a bubble chamber, courtesy of Fermilab; and the twisting of spacetime due to a spinning mass, courtesy of the Gravity Probe B collaboration.)

 

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.

Noether’s Theorem

Informally, W.J. Thompson writes Noether’s theorem as:

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.

diaond rotation
If we rotate a square (left) 45 degrees counterclockwise, it becomes a diamond (right).

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.

Rotating a square by ninety degrees leaves it unchanged.
If we rotate a square by ninety degrees, it appears unchanged. This is called a “symmetry” of the square.

Incidentally this would remain true even if we rotated the square by 2\times 90 = 180 degrees or by 3\times 90 = 270 degrees or by 4\times 90 = 360 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. 1+3 = 3 + 1 = 4. 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.

Order matters in symmetry
Order matters in symmetry. If I rotate a cube ninety degrees towards myself and then ninety degrees clockwise (top), I get a different orientation (shown by the differently colored faces of the cube) than when I rotate a cube ninety degrees clockwise and then ninety degrees towards myself (bottom).

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 72 degrees, 2\times 72 =144 degrees, 3\times 72 = 216 degrees, 4\times 72 = 288 degrees, and 5\times 72 = 360 degrees.

Let’s add another side. A hexagon is symmetric under rotation by 60 degrees, 2\times 60 = 120 degrees, 3\times 60 = 180 degrees, 4\times 60 = 240 degrees, 5\times 60 = 300 degrees, and 6\times 60 = 360 degrees.

A pentagon is symmetric under rotation by 72 degrees. A hexagon is symmetric under rotation by 60 degrees.
A pentagon is symmetric under rotation by seventy-two degrees. A hexagon is symmetric under rotation by sixty 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!

Symmetries in Three Dimensions
In three dimensions, a paraboloid (left) is symmetric under rotation about its axis, a sphere (center) is symmetric under rotation about its center, and a plane (right) is symmetric under motion in a straight line.

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.

A ball in a paraboloid with no angular momentum
If we drop the ball straight into the paraboloid, it just oscillates left and right and up and down. (Part of the paraboloid cut out for clarity. The blue arrow shows where we dropped the ball and with what initial velocity.)

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:

If we drop the marble into the paraboloid at a slight angle, rather than straight down, then it travels around the axis of the paraboloid.
If we drop the marble into the paraboloid at a slight angle, rather than straight down, then it travels around the axis of 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.

Implications

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.

Further Reading

  • The New York Times did an article on Emmy Noether here.
  • The Examiner also did an article on Noether here.
  • The University of California at Las Angeles has some records of her here.
  • Theoretical physicis John Baez has an article on Noether’s theorem for the interested physics student.
  • Professor Nina Byers of UCLA, details the story of Noether’s discovery of her theorem here.

Questions? Comments? Insults?

As always, if you have any questions or corrections, or if you just want to say hi, please leave a comment or shoot me an email.

Posted in abstract algebra, History, Mathematics, Physics, Quantum Mechanics, Relativity, Science And Math | Tagged , , , , , , , , , , | 2 Comments

How Things Work: Lasers

You know,
I have one simple request.
And that is to have sharks
with frikkin laser beams
attached to their heads!

~Dr. Evil

Always look on the bright side
…unless you’re holding a laser pointing device.

~Unknown

Pew pew pew!
The death star superlaser embodies our fascination with lasers. But how do they work? (Source: Death Star PR)

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.

An electron around an atom
If an electron’s wavelength is too short (left) or too long (right), then it doesn’t fit at a given radius. However if the wavelength is an integer value of some special number (center), the electron fits.

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.

Energy level diagram
An energy level diagram. The red bars represent energies and momenta that the electrons in an atom are allowed to have. The higher up the bar is, the more energy. The further to the right the bar is, the more momentum.

Between Light And Matter

 Now let’s imagine an electron sits in the lowest energy level, as shown below.

electron just sitting. yo.
An electron (yellow) sitting in the lowest energy band (red bar).

When a photona 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.

Electron absorbs a photon. Or not.
Left: A photon with the right energy and momentum (green) hits the atom, causing the electron (yellow) to absorb the photon and jump to a new allowed energy band.
Right: a photon with the wrong energy and momentum (blue) hits the atom. The electron (yellow) is unable to absorb the photon because the electron’s new energy and momentum would not be allowed. The photon passes through the atom unmolested.

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.

Electron emits light
If an electron (yellow) is in a high-energy state, it will try to lose energy by emitting a photon (green) and drop to a lower-energy state. The photon then will have the energy and momentum that the electron lost.

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.

photon bombardment
Left: If we bombard an electron with photons with the wrong energy and momentum (blue), eventually one of the correct energy and momentum (green) will come along, as shown in the right, and excite the electron.

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.

Quantum electron fluctuates out of quantum allowed state.
Because of the uncertainty in its energy and momentum allowed by the Heisenberg uncertainty principle, the electron (yellow) can sometimes, very rarely, fluctuate out of its allowed energy states, at which point it can absorb a forbidden photon (blue) if the photon puts it into an allowed state.

Stimulated Emission

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.

Stimulated emission
When an electron is in an excited state, an incident photon of the appropriate energy and momentum can cause the electron to emit another photon which is an exact clone of the incident one. This is called stimulated emission.

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.”

The lifecycle of a laser.
The lifecycle of a laser. An electron starts in a low-energy state (lower left), but is excited by some process to a high energy state. Then the electron quickly relaxes into a middle-energy state (top). The lost energy might go into the vibration of the atom or into light we don’t care about. The electron then sits in this middle energy state for a long time, allowing us to create a population inversion. With a population inversion, we can induce stimulated emission (right).

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.

A laser cavity.
A so-called laser cavity, where we place the gain medium between two mirrors.

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.

Resonance

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.

This is also called a Fabry-Perot Cavity
Light waves in a laser cavity. The longest wavelength wave is the blue. The shortest is the red. (This is actually the opposite color coding as that in real life. Sorry about that.) If these waves had slightly longer or shorter wavelengths, they wouldn’t fit in the cavity.

This selection process is an example of a broader phenomenon called resonance. The mythbusters have a nice explanation of resonance in their episode on Tesla’s earthquake machine.

Applications?

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.

Further Reading

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 National Ignition Facility, where they’re trying to use lasers to make fusion power has a nice article.
  • How Stuff Works has an article on lasers too.
  • LFI International has a nice article too.

Questions? Comments Insults?

As always, let me know if you have questions, comments, or hatemail. Or if you just want to speak your mind.

Posted in optics, Physics, Quantum Mechanics, Science And Math | Tagged , , , , , , , | 2 Comments

Why Black Holes Glow: Accretion Disks

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.

~Michael Dirda

Galactic center
A radio image of the center of the galaxy. The bright glow in the center is partly due to the super-massive black hole, Sagittarius A*. (Source).

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.)

Accretion disk
We think of black holes as, well, black. However, many of them are the brightest objects we see in the sky. This simulation of a black hole reveals why: Black holes are surrounded by glowing matter, called accretion disks. (Source: NASA)

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.

Centrifugal Force

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.

Of course, we know now that Newton and Huygens were both right (to a degree). Quantum mechanics has shown us that light is both a particle and a wave. It bends and refracts like a wave, but it carries energy and momentum like a particle. This means it can bounce off of things and exert force. (Although light is a wave, it doesn’t need a medium like sound does. It can propagate in empty space.)

Photon has self-identity problems
We know from quantum mechanics that light has both a particle nature and a wave nature. (Source)

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.

mirror mirror on the wall...
We can think of light waves bouncing off of a mirror (left) as a stream of particles (right). Since the mirror is pushing on the particles, they push back, exerting a force on the mirror towards the right.

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.

Solar sail NASA
A solar sail is a gigantic mirror off of which we bounce light from the sun. If the sail is big enough, the force exerted by the photons will be enough to move a spaceship. (Image courtesy of NASA.)

(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.)

Jets

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.

Further Reading

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.

Questions? Comments? Insults?

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!

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

Between the Two Shores: Covalent Bonding

But let there be spaces in your togetherness
and let the winds of the heavens dance between you.
Love one another but make not a bond of love:
let it rather be a moving sea
between the shores of your souls.

~Kahlil Gibran

team building sphere
Are the quantum covalent atomic bonds anything like the bonds between human beings? (Source: Pink Potato Team Building)

Two weeks ago now, I flew to Conway, Arkansas to attend the wedding of my very good friends Vincent and Mary. This and an academic conference got in the way of blogging for a little while but I’m back. As such I decided to a post in their honor about bonding. Not human bonding, mind you, but on chemical bonding. Specifically, covalent bonding! You probably know that atoms missing electrons like to form covalent bonds with each other where they share their electrons. But why does this happen? The secret lies in the quantum mechanical nature of electrons.

This post will rely heavily on the articles I’ve previously written about quantum mechanics. You might want to check out my previous posts. These ones will be particularly helpful:

If you don’t want to go through all of these, I will try and link to the relevant ones as they come up in discussion.

Energy

In physics, we have this thing we call energy. We usually break energy into two categories: kinetic energy and potential energy. (There are more “types,” but in the end, they can be reduced to these two types.)  Kinetic energy is a little easier to understand, so let’s talk about that first.

Roughly, Kinetic energy measures how much something moves, and how difficult it is to make that thing move or to stop it once it’s moving. In classical mechanics, the kinetic energy is given by one half the square of the velocity times the mass,

    \[KE = \frac{1}{2} m v^2.\]

(Astute readers may remember that I described momentum in a similar way. I said it measured how much something is moving and how difficult it is to change an object’s motion. Kinetic energy and momentum are very much related. The biggest difference is that momentum is a vector. It has a size and a direction… and it measures the direction of motion as well as the resistance to change. On the other hand, energy is a scalar. It’s just a number, which comes from squaring the velocity. Also, although energy can be transferred between objects and transformed between potential energy and kinetic energy, momentum only describes motion. Finally, as  a rough intuition, momentum measures how difficult it is to make a small change in an object’s motion, while kinetic energy measures how many small changes are required to change the motion in a big way.)

Potential energy measures the ability to generate kinetic energy. If I’m very high up on a cliff and I jump off, I can accelerate very fast and acquire a lot of kinetic energy (which I stole from the Earth’s gravitational field). This means that my potential energy (at least from gravity) is proportional to my height. Other sources of potential energy include electric and magnetic fields, springs, and even massive particles themselves.

kinetic vs potential energy
The story of how I get kinetic and potential energy if I climb up a cliff and jump off. Don’t worry, I have a parachute.

The total energy of a system or a particle is the sum of the kinetic energy and the potential energy. And the total amount all energy in the (classical) universe is conserved; It can’t be created or destroyed, simply passed around between particles, objects, and people.

(Experts know I’m glossing over a lot here. In truth, the distinction between kinetic and potential energies is pretty artificial. The physicist and mathematician Emmy Noether defined energy as whatever quantity a physical system possess that doesn’t change in time. In other words, it is the time-translational symmetry of the system. We’ve simply given names to the contributions to the energy like kinetic and potential energy. And indeed, energy may not be conserved for the universe as a whole. In Einstein’s general relativity, energy is not necessarily conserved.)

Of Energy and Wiggles

I’ve described what energy means in a “classical” system, where quantum effects are negligible. But in quantum mechanics, things get a bit weird (as they often do). If we understand kinetic energy, it’s simple enough to define potential energy as the ability to create kinetic energy. But what does kinetic energy mean in the quantum case? We defined it as the motion of a particle, but quantum particles don’t travel in the same way… they’re waves that exist everywhere at once. What does motion mean in this context?

We can take a hint from the wave nature of quantum particles. In quantum mechanics, the height of the wave at a given position tells us how likely it is we’ll observe a particle there. But all waves wiggle. The height rises and falls. And it just so happens that the number of times the wave wiggles over a given distance determines the energy! I’ve plotted three quantum probability functions below, each with a different energy. In this case, the quantum particle—say an electron—is confined to a box of length 1 by an infinitely strong electric field. This means that the probability of measuring the particle outside of the box is zero, and the height of the wavefunction must reflect that.

waves in a box
Three quantum particles confined in a box of length 1. The one with the fewest wiggles (green) has the lowest energy. The one with the most wiggles (blue) has the most energy.

Now the thing is, the energy depends on the wiggles per unit volume. So if we made the box longer, all three particles would lose energy because they’d all still have the same number of wiggles, but there would be fewer wiggles per cubic meter. Let’s make a note of that. It’ll be important later.

The Electron and The Nucleus

In the past, I’ve given you Niels Bohr’s description of the atom, which demonstrates that electrons in atoms can only have specific kinetic energies. This is because only an integer number of wiggles fit around a circle if you want the wave of a particle to agree with itself after you go 360 degrees around the circle. The same holds true for most quantum systems.

For simplicity, lets look at the hydrogen atom in a different light, though. Let’s imagine that a hydrogen nucleus (i.e., a proton) exerts an attractive force on the electron and ignore that the electron can orbit around the nucleus. In other words, let’s imagine one-dimensional atoms.

Because the proton is so much heavier than the electron, we can basically think of it as stationary. The attractive force between the electron and the proton gives the electron some potential energy depending on its position in space. To make the whole problem easier to visualize, let’s make an analogy with gravity. On Earth, when we’re high up, we have a lot of potential energy and when we’re down low we have very little. We can describe the potential energy of the electron by plotting as if it were a height above the ground (or a depth below the ground). In the case of the hydrogen atom, that potential energy looks something like this:

single proton potential
The potential energy of a single proton.

In our little approximation, the electron is more strongly attracted to the proton the closer it gets to the proton. If it overlapped with the proton, it’d be infinitely attracted to it. However, particles in quantum mechanics have a minimum energy they’re allowed to have. In this case, that minimum is the Bohr energy. If we plot the probability distribution for an electron in the lowest energy state of the atom, it looks something like this:

Electron in atom
A single quantum-mechanical electron in a hydrogen atom. The red line is the potential energy of the electron. The blue line is the probability density wavefunction and the green line is just an axis to show zero.

If the electron were classical, it couldn’t go farther away from the center of the atom than the classical turning points, which I’ve marked with big black dots. This is because the electron doesn’t have enough energy to “climb” out of the potential energy well and leave. But in quantum mechanics, the electron can exist in places it’s classically forbidden to be. This is very similar to quantum tunneling. This uniquely quantum behavior is critical to explaining how atomic bonds work.

One Electron, Two Nuclei

Imagine we take our hydrogen atom and move it next to a proton. Now there are two potential wells like the one above. If the wells are far enough apart, the electron only sees its own proton, it’s own hydrogen atom, as shown below.

electron before bonding
An electron in a hydrogen atom near a proton. The electron doesn’t really notice that it could have a low potential energy (red) by going to the proton… they’re too far away. So the probability density function (blue) looks like it did before.

However, if we move the new proton close enough to the hydrogen atom, the potential energy profile changes. It starts to look something like the figure below.

empty double well potential
When we bring the second proton close enough to the hydrogen atom, the potential energy profile changes and less energetic electrons can exist between the two protons (the nucleus and the extra proton).

And now the true quantum nature of the electron comes into play. Remember when I said that quantum particles can exist where they classically should not? Well when we bring the two “potential wells” together, the electron in the left well has some probability of existing between the two wells. And, indeed, even if it started in the left well, the electron will ooze into the right well so that it spends about half its time in each well. Then the picture of the wavefunction look something like this:

Two nuclei, one electron.
The wavefunction of a particle (blue) living between two hydrogen nuclei. The potential energy induced by the nuclei is shown in red. Even though the electron may have started in one of the two nuclei, it has oozed into the other nucleus. This means it has spread out and is in a lower energy state than when there was only one nucleus.

But now something funny has happened. The electron used to have one wiggle in some amount of space. Now it has more wiggles, but it also has a lot more space. The result? The electron has lost a lot of energy!

This is why atoms bond. The electrons in an an atom want to be in the lowest energy state they can, and adding another atom lets the electrons lose some energy. There’s an optimal distance between the two nuclei which gives the electron a minimal energy. And this is what controls the length of atomic bonds.

Two Electrons?

Usually each atom in a covalent bond has an electron, not just one of the atoms. Fortunately, so long as there are only two electrons in the shared orbit (the one where the bonding happens), the electrons don’t see each other at all. Each electron chooses a “spin,” which has to do with the magnetic field an electron produces. (Spin will be the subject of a later article, I promise.) There are two possible spins and, so long as each electron has a different spin, they don’t see each other. However, if more electrons appear, we get a problem because there are only two possible spins and the third electron must choose a spin that’s already been taken. Then the electrons repel and the bond breaks.

See for Yourself?

The physics education group at the University of Colorado at Boulder has developed a simulation of a quantum particle bound by two potential wells. Click on the image below to see it in action! For the atomic bonding case, change the toggle on the right from “square” to “1D coulomb.”

Double Wells and Covalent Bonds

Click to Run

Questions? Comments? Insults?

This post is a bit technical so if you have any questions please do ask! And if you’re a physical chemist and you know better than me, pipe up!

Posted in Physics, Quantum Mechanics, Science And Math | Tagged , , , , , , | 1 Comment