In 1924, Louis de Broglie published his doctoral thesis in which he put forward a hypothesis today known as "de Broglie's relation". This was one of the integral pieces of the early development of quantum mechanics and earned him the Nobel prize in physics in 1929. The relation says that any particle with momentum p will propagate as though it were a wave of wavelength

λ = h/p

where λ is known as the de Broglie wavelength. Often amongst physicists, it is written in the more convenient form

p = hbar k

where hbar is Dirac's constant (h/(2π)) and k is the wavenumber. It may seem strange that matter would be expected to behave in a wave-like fashion, but this has been observed in experiment (discussed below). While it was quickly clear that particles exhibited wave-like properties, it wasn't immediately clear what exactly the waves meant or what they were propagating in. The quest to answer those questions became known as the interpretation of quantum mechanics, with the Born statistical interpretation of the wavefunction (a wavefunction is the mathematical representation of a wave) being the first major advance in understanding the meaning of de Broglie waves. The statistical interpretation says that the squared amplitude of the wave at a certain location tells you the probability of finding the particle at that location. The fact that particles have both wave-like and particle-like properties is known as the wave-particle duality. Not long after de Broglie made his hypothesis, Schrödinger tried to develop a wave equation that would have de Broglie's relation as its dispersion relation, and he came up with Schrödinger's Equation.

Experimental Evidence

There are many experiments that demonstrate the wave nature of matter. One early experiment of this type was the 1927 experiment of Davisson and Germer in which electrons hitting a nickel crystal exhibited a diffraction pattern. This sort of effect is predicted by wave theories but is not expected for classical particles. Further experiments by Thompson, Rupp, and others confirmed the existence of these effects. Later in 1961, Claus Jönsson demonstrated the same effect in a very clear way by doing Young's double slit experiment with electrons. Though de Broglie's hypothesis seemed pretty radical when he first made it, experimental evidence quickly forced physicists to take it seriously.

Wave Packets, Fourier Transforms, and the Uncertainty Principle

de Broglie's relation gives us new relationships between position and momentum that we did not have in classical mechanics. A particle with one momentum will have a sinusoidal wavefunction of a certain wavelength according to the relation, meaning that it will be spread out all over space. In order to get a particle that's localized (i.e. very likely to be in a fairly small region of space), we want a wave packet, a wave form that is only significantly different from zero in a small area. Fourier analysis tells us that we can make a wave packet by adding together waves of several different wavelengths in a superposition. By making the right combination we can use interference to cancel out most of the original wave leaving only a small "wave packet" in one place. In order to get a smaller wave packet we must add more and more wavelengths to make the cancellation more exact. de Broglie's relation tells us that a particle with such a wave packet would have a combination of several different momenta. What does that mean exactly? Well, the statistical interpretation says that that superposition means that if you measure the momentum you could get any one of the values that make up the wave packet.

So, from the properties of waves we have discussed, we can say that a particle having a single, well-defined, momentum will be spread out over many possible positions, while a particles whose wave packet describes a fairly well localized position will have many possible momenta. One may make this idea more precise mathematically by defining the momentum wavefunction, which is the Fourier transform of the position wavefunction and gives the probability of measuring each momentum for that state. From this mathematical framework one may derive the Heisenberg Uncertainty Principle, which then follows simply from known results in Fourier analysis.

Wavelengths and Length Scales

It is a widely applicable principle in wave mechanics that you will only start to see "wave features", like diffraction and interference, when you start looking phenomena happening on distance scales similar to (or smaller than) the wavelength. This is precisely the reason why we can understand a lot about light thinking about it as "rays", even though it is actually a wave. The same applies for "matter waves" that behave according to de Broglie's relation. If you consider the behavior of these waves going through openings much larger than one wavelength or in general if you look at the patterns with a device that can't resolve details as small as a wavelength, then you can understand the behavior of the wave in terms of rays or particles flying along straight paths. This has two different implications. The first is that a macroscopic object with even a little momentum will have a very small de Broglie wavelength, so we would not be able to notice any wave behavior of the object without very precise instruments^*. The other implication is that if we want to bounce particles off of an object to determine its structure, we will have to use higher momentum particles in order to see smaller scale structure, so physicists usually draw a connection between high momentum and short length scale.

---

* This argument is a bit of a swindle. There's actually more to it than this, because of course you could ask how we know a macroscopic object will never have small enough momentum to become delocalized. To really get a satisfactory answer you must turn to quantum decoherence.

Sources

• The Physical Principles of the Quantum Theory, Werner Heisenberg
• Introduction to Quantum Mechanics, David J. Griffiths
• Clinton Davisson's Nobel Lecture, http://nobelprize.org/physics/laureates/1937/davisson-lecture.pdf
• http://www.slac.stanford.edu/library/nobel/nobel1929.html

The Millikan Oil Drop Experiment measures the charge of the electron. Designed and first performed by Robert Andrews Millikan of Caltech in 1909, the experiment shows that the oil droplets are charged in integer multiples of -1.6*10^-19 Coulombs. This proved that charge was quantized, and showed the magnitude of that quantum. Millikan won the Nobel Prize in Physics in 1923 for this work.

The experiment works as follows:

Millikan put oil in an atomizer, much like a perfume bottle, and put it over a chamber with two parallel plates, top and bottom, and a hole in the top one. There was a microscope looking into the side of the chamber. First, he measured the terminal velocity of the oil drops without any charge, which allowed him to calculate the mass of the oil droplets.

Then, by applying X-rays before the droplets enter the chamber, he was able to add unknown amounts of charge to each drop. He would then vary the voltage on the two parallel plates until a given droplet hung still in the air. He knew the forces worked out thusly:

^
| (charge of the drop * magnitude of the electric field)
|
Droplet
|
| (Mass of the droplet * the acceleration of gravity on Earth, 9.8 meters/second^2)
V

Since those two forces had to balance each other out, and mass, gravity, and the electric field are known, it is a trivial matter to calculate charge. From there, one just has to calculate the Least Common Multiple, and that is the value of charge for e, the electron.

The website below has a simulator for the experiment, which is neat. Requires a Java Virtual Machine.

Sources: high school AP Physics course, and http://www68.pair.com/willisb/millikan/experiment.html for a refresher.

Commonly abbreviated as RISUG (pronounced "RICE-ugh"), reversible inhibition of sperm under guidance, is a safe, long-term, reversible, non-hormonal method of male birth control in which a copolymer of Styrene malic anhyrdride (SMA) with Dimethyl sulfoxide (DMSO) is injected into the vas deferens. The copolymer forms a matrix across the vas which allows sperms to pass, but which carries electric charge which ruptures the acrosomal membrane of the sperms, rendering them immotile. RISUG has two main advantages over classical occlusion methods: first, it is easily reversible, requiring only an injection of DMSO or a sodium bicarbonate solution to dissolve the SMA matrix, and it does not cause an immune response against sperm prevented from leaving the body (and thereby does not prevent the production of viable sperm after the reversal of the procedure). Second, RISUG is almost entirely noninvasive-the procedure involves a small injection and is effective within hours. Like traditional occlusion methods, RISUG is 100% effective, Furthermore, RISUG is safe; in eleven years of human testing, RISUG has caused no long term side effects {2}.

Unlike hormonal methods of birth control, which affect the basic chemical processes in our bodies, RISUG has no potential to affect human behavior. And, unlike condoms, RISUG has the convenience of not requiring any specific preparation prior to sex. RISUG has great potential, especially in the United States of America, where there is a (false) stereotype that men do not wish to (or are unable to) take responsibility for sex. Currently, the only available male birth control technique is the condom, which is inadequate in success rate (it has a 15% failure rate for what Planned Parenthood calls "typical use"), convenience and, many feel, intimacy{3}. Hormonal techniques may prove effective {4}, but many would never use a male version of the birth control pill: many men simply do not want to subject their bodies to artificial versions of our own hormones-chemicals that affect us in extremely basic and powerful ways.

Not only would RISUG give men a safe, reliable, convenient method of birth control, so that we might share responsibility for sex with our lovers, but it would allow women in relationships with men to, if they desire, discontinue the use of birth control pills which may adversely affect their health and or behavior (for example, many women are thrown into heavy depression by most or all birth control pills).

Currently, RISUG is in Phase III testing in India (where the government is always in favor of more and better birth control techniques). With luck, RISUG will soon by publicly available there {1}.

RISUG may never be available in the United States, however. It does not pass Western standards for pharmaceuticals, in that it isn't sufficiently profitable. Women spend thousands of dollars a year on birth control. Men might pay $500 for an injection of RISUG which would last for ten years {1}. Furthermore, it would require a large investment of money and time to perform the testing required for FDA approval. A company that wanted to finance the testing of RISUG in the United States would be taking a substantial risk-the drug might not recoup expenditures fast enough to warrant the spending. That something as beneficial to the public good as RISUG may be overlooked in America because it isn't profitable is a sure sign that pure capitalism is not a good system to run a country with (yes, I am well aware that the US has a mixed economy).

Sources:
1. Methods - vas devices - RISUG. www.malecontraceptives.org
2. United States Patent: 5,488,075; Contraceptive for use by a male. www.uspto.gov Patent Database
3. Planned Parenthood - The Condom. www.ppfa.org/bc/condom.htm
4. Planned Parenthood - The Future of Birth Control. www.ppfa.org/ARTICLES/bcfuture.html

What makes atmospheric effects so cool to me is that, if I was sitting around the universe, bored, and on a whim decided to come up with a moist, rocky planet in a nitrogen-oxygen-based atmosphere around a sun, I'm not sure I could have predicted them. Yet they are some of the most awe-inspiring and eye-filling visual experiences you can have.

Atmospheric effects is the collective term of optical phenomenon observed in the sky and so are also called atmospheric optics, meteorological optics, and aerial spectra. Atmospheric effects include mirages, rays, shadows, and reflective and refractive effects seen in rainbows, aureoles, halos and arcs.

Atmospheric optics is a subset of the atmospheric sciences. As a field it is much more concerned with explanation and observation than history, and very little information is available about its forbearers. Listed below are milestones of optics history that deal specifically with atmospheric effects.

Brief History: Aristotle to Newton

• In book 3 of his Meteorologica (circa 350 BCE), Aristotle discussed many of the phenomena noded here, though he erroneously attributed them all to reflection. For two millennia this work was regarded in the west as the authority in meteorology.
• Following Aristotle closely, Alexander of Aphrodisias took his hand at describing some atmospheric phenomena in 200 BCE. Of particular note is his observation of the dark band that appears between double rainbows and that bears his name today.
• A disciple of Plato, Philippus of Opus (4th century CE), wrote many books. One of the fragments that survive discusses the rainbow as a phenomenon of diffraction.
• In the 9th century CE, the philosopher-of-the-Arabs Jakub Ibn Ishak Al Kindi became enamored with the Greek texts (again, Aristotle) as he was helping to translate them. He wrote a book titled Optics that included some atmospheric effects.
• Shortly after Al Kindi, Ibn Al-Haytham (aka Alhazen and al-Haitham), another Arabic natural philosopher in Basra, wrote 14 books including Kitab al-manazir (The Book of Optics), and Maqala fi qwa' quzah wa al-hala (Treatise on the rainbow and the halo). He also wrote a lost book dealing with the colors of the sunset.
• In the earlier part of the 14th century CE, Theodoric of Freiberg correctly explained rainbows as a consequence of refraction and internal reflection within individual raindrops. However, he incorrectly explained chromatic aberration.
• In the early 17th century CE the Croatian scientist Marko Dominis discussed the rainbow in De Radiis Visus et Lucis (On the Rays of Sight and Light in Lenses and the Rainbow) where he correctly explained the inner arc of the rainbow.
• In the mid 17th century CE, Rene Descartes quantitatively derived the angles at which primary and secondary rainbows are seen with respect to the angle of the sun's elevation.
• In the early 18th century CE, Isaac Newton published Opticks, in which he correctly explained diffraction and chromatic aberration.

Halo Displays

When five or more types of atmospheric effects are visible at once, the results are spectacular and breathtaking. Since such displays almost always involve at least one halo, they are called halo displays, or halo phenomenon. It's worth noting how many of the halo displays create shapes that resemble symbols with deep human spiritual significance:

• 22° halo + parhelia +sun pillars: Greek Cross
  Emperor Constantine I of the Roman Empire is said to have seen such a display in 313 CE near Trier. This sign is supposed to have prompted him to become a Christian.
• Heliac arc: Ichthus (∝)
• Tricker + diffuse arc: Ankh
• 46° halo + circumzenith arc: Crown with a crescent moon
• Wegener + diffuse arcs + invisible Earth: (theoretical, see the node) Lemniscate (∞)

Mark Vornhusen with Germany's AKM shows evidence that there are halo displays described in the Bible (St.John of Jerusalem´s revelation, Daniel's and Ezekiel's visions) and even that many of Hildegard von Bingen's 36 visions of heavenly figures were probably halo displays.

Predicting and Studying

It used to be that atmospheric effects science was a matter of patient observation, documentation, and debate. Nowadays, rather than wait around for good fortune and mother nature to get their respective acts together and make one of these things happen, computer science has aided the study of atmospheric effects by making it primarily a mathematical modeling problem that seeks to confirm hypotheses through comparison to observed facts. Scientists use raytracing programs to model different particles at different angles and dispersions with varying light sources. This software is not that computationally intensive and can run on modern (2004) desktops. In fact, all of the following are available for free download.

• BowSim models rainbows.
  http://www.sundog.clara.co.uk/rainbows/bowsim.htm
• HaloSim models halo effects.
  At http://www.sundog.clara.co.uk/halo/halfeat.htm
• IRIS simulation models glories, fogbows, and aureole.
  http://www.sundog.clara.co.uk/droplets/iris.htm

Other worlds

Our Terran atmospheric effects-especially the ice-based effects-look the way they do partially because of the shape, temperature, and composition of our atmosphere, but mostly because of the molecular structure of water. On other worlds where atmospheric crystals are made of other chemicals, these effects would look different. Altering the modeling software for other crystal shapes, optics can predict the appearance of these extraterrestrial halos and arcs. For example, the octahedral ammonia crystals in the atmospheres of our gas giant planets produce four sundogs instead of two. The Atmospheric Optics site has fine examples of Mars, Jupter, and Saturn.

Where can you see atmospheric effects?

Water-based refraction effects are common all over the world. The other effects are harder to come by. Les Crowley summarizes the perfect formula as follows:

The recipe for an extraordinary display is beguilingly simple. Take a clear sky, cover it with a thin and uniform cirrus haze. Be sure to populate the haze with large and near optically perfect ice crystals of many varieties and precise orientations.

Alternatively in very cold weather, fill a clear blue sky with equally perfect low level diamond dust crystals.

Where would this happen? Some of the most spectacular photographs come from the poles. But as most of us aren't Naomi Uemura, it stands to reason that the closer you get to the poles, the more likely you are to run across these. Russia, Scandinavia, Alaska, Canada and Antarctica seem likely locations if you're trying to stack the deck, but many of the effects are visible at lower latitudes, too. I've seen several since I moved to Seattle. One thing you can do where you are is to simply get in the habit of looking for them. Many of the common halo effects go unnoticed simply because people don't glance up, or recognize what they're looking at.

If you do happen to notice one of these effects, remember that they seldom last longer than an hour. It might be worth stopping what you're doing to appreciate them while they're there. I've found that they are an excellent moment to recontextualize your troubles. A memento belli of sorts.

---

_Sources
• _{http://encarta.msn.com/encyclopedia_761571037_4/Meteorology.html#s34}
• _{http://www.sundog.clara.co.uk/atoptics/phenom.htm}
• _{http://www.meteoros.de/indexe.htm}
• _{http://amsglossary.allenpress.com/glossary}
• _{http://homepages.wmich.edu/~korista/atmospheric_optics.pdf}
• _{http://www.nationmaster.com/encyclopedia/Timeline-of-electromagnetism-and-classical-optics}
• _{http://www.sciencedaily.com/releases/2000/03/000314065455.htm}
• _{And one charming email from Les Crowley, inquiring after historical sources:
  > Yes you have a point. I have stated the current position rather than use a historical approach.
  > There are many individual bits of history on the excellent AKM site but a single source does not spring readily to mind, I will think about it while doing some painting!
  > Les}

Hydrostatic equilibrium is one of the most important fundamental principles in atmospheric physics and astrophysics. It defines the properties of stable gaseous systems confined within a gravitational field, and can be used to estimate the properties of Earth's atmosphere, Jovian planets, stars, and even clusters of galaxies. In its simplest form, it states that the inward force of gravity on an infinitesimal parcel of gas is balanced by the outward force of pressure by the gas underneath it.

The simplest physical case of an isothermal (uniform temperature), ideal gas is the most physically enlightening.

Take a cube of gas, with sides dx=dy=dz and density ρ, bounded top and bottom by gas pressures P₁ and P₀ at a height z in a (nearly) constant gravitational field.

              P(1)
         __________ 
        /         /|
       /         / |
      /         /  |dz  gravity (g)
     |---------|   |       |
     |         |   |       V
     |         |  /
     |         | / dy
     |_________|/
           dx

            P(0)

The sides of the cube have area dA = dxdy. We assume that gravity is only acting in the downwards direction, and there are no lateral accelerations, so we ignore motions of the cube in the X and Y directions. For the cube to be static, all vertical forces on the cube must sum to zero. The net downwards force is the force of gravity on the cube plus the downwards pressure from gas above the cube, or

F_down = - M_cube g - P₁ dA (eq. 1)

Note the minus signs, indicating that the force is acting in the negative (downwards) direction along the z-axis. The net upwards force is just the pressure of the gas below the cube, P₀ dA. The sum of the forces is then

F_down + F_up = -M_cube g + (P₀ - P₁) dA = 0 (eq. 2)

Since M_cube is just ρ dx dy dz = ρ dz dA, the equation reduces to

-ρg dz = P₁ - P₀ = dP (eq. 3)

Now, since we're talking about an ideal gas, P = ρ k T / μ, where k is Boltzmann's constant, T is the temperature, and μ is the mean molecular weight. We can substitute this into equation (3) and get

P (-μg)/(kT) dz = dP (eq. 4)

Integration of this equation yields

ln(P) + const = -(μg)/(kT) z (eq. 5)

where ln is the natural logarithm, resulting from the integration of dP/P. This is then usually written in exponential form, as

P = P(z₀) exp(-μgz/(kT)) (eq. 6)

where P(z₀) is the pressure at the (arbitrary) point z₀ (on Earth, this would be the equivalent of sea level). The quantity (kT/μg) is usually rewritten as H, the scale height, leading to the simple equation

P = P(z₀) exp (-z/H) (eq. 7)

On Earth, the value of the scale height is about 8 km, meaning that the air pressure drops by a factor of e (about 36 percent) when you increase your altitude by one scale height.

Obviously, there are some caveats to using the above formulation. First, the local acceleration due to gravity, g, is not a constant, so to be precise, it should be replaced with Newton's Law of Gravitation:

g = GM/R² (eq. 8)

Second, the gas you deal with will be neither isothermal nor ideal, so they introduce a further difficulty into the equation. However, as written, equation (7) provides a good, first-order approximation to the behavior of gaseous atmospheres. If you happen to be an astronomy or atmospheric physics grad student, it will almost certainly appear on an exam at some point so this derivation is worth knowing.