Advent
drorzel | December 24, 2011As we started the last week of the advent calendar, I was trying to map out the final days, and was coming up one equation short. I was running through various possibilities-- the Dirac equation, Feynman's path integrals, the Standard Model Lagrangian, when I realized that the answer was staring me right in the face:
This is, of course, the Heisenberg Uncertainty Principle, saying that the product of the uncertainties in position and momentum has to be greater than some minimum value. Strictly speaking, this should've come before the Schrödinger equation, if we were holding to chronological…
drorzel | December 23, 2011A week and a half ago, when the advent calendar reached Newton's Law of Universal Gravitation, I said that it was the first equation we had seen that wasn't completely correct. Having done our quick swing through quantum physics, the time has come to correct that equation:
If you say "Einstein equation" to a random person on the street, odds are they'll immediatley think of "E=mc2." If you ask a physicist to think of the Einstein equation, though, this is the one they'll think of. This is the Einstein field equation from general relativity, and while it's not as well known as E=mc2, it's…
drorzel | December 22, 2011Newton's birthday (in the Julian calendar) is Sunday, so we're in the final days of the advent calendar. Which means it's time for the equations that are least like anything Newton did, such as today's:
This is the Schrödinger equation from non-relativistic quantum mechanics. If you want to determine the quantum state of an object that's moving relatively slowly, this is the equation you would use.
It also has probably the greatest origin story of any of the equations we've talked about. Or at least the most salacious origin story of any of the equations we've talked about...
Erwin…
drorzel | December 21, 2011Today's equation in our march to Newton's birthday is actually a tiny bit out of order, historically speaking:
This is the Rydberg formula for the wavelengths of the spectral lines in hydrogen (and hydrogen-like ions), with R a constant having the appropriate units, and the two n's being two dimensionless integers. This equation was developed in 1888 by the Swedish physicist Johannes Rydberg (who was generalizing from a formula for the visible lines of hydrogen that was worked out by a Swiss schoolteacher, Johann Balmer). As such, it pre-dates Einstein's equation from yesterday, but its…
drorzel | December 20, 2011Yesterday's equation was the first real result of quantum theory, Max Planck's formula for the black-body spectrum. Planck never really liked the quantum basis of it, though, and preferred to think of it as just a calculational trick. It wasn't until 1905 that anybody took the idea really seriously, leading to today's equation:
From the year, you can probably guess the guy responsible: Albert Einstein. Einstein realized that if you took Planck's idea and ran with it, you could explain the photoelectric effect very neatly. Where Planck had viewed the quantized radiation as a fictitious…
drorzel | December 19, 2011Moving along in our countdown to Newton's birthday, we come to 1900, and one of the most revolutionary moment in the history of physics, represented in today's equation:
This is Max Planck's formula for the spectrum of the "black-body" radiation emitted by a hot object at temperature T. It's also the equation highlighted on what might be the most famous xkcd cartoon (albeit in different notation).
This is a fitting next step in the countdown not only for reasons of chronology, but also because it's a nice bridge from thermodynamics and statistical mechanics. After all, the red glow of a hot…
drorzel | December 18, 2011As I said yesterday, I'm going to blow through another entire subfield of physics in a single equation, as our march toward Newton's Birthday continues. Today, it's statistical mechanics, a very rich field of study that we're boiling down to a single equation:
This is Boltzmann's formula for the entropy of a macroscopic system of particles, which says that the entropy s is proportional to the logarithm of the number N of microscopic states consistent with that macroscopic state. The constant kB is there to get the units right.
Why does this get to stand in for the whole field of statistical…
drorzel | December 17, 2011Once again, the advent calendar is delayed until late at night by a busy day with SteelyKid-- soccer in the morning, playing with a trebuchet after lunch, then Arthur Christmas at the Colonie mall. We're running low on days to honor great milestones in physics, though, so I don't want to skip a day entirely.
I'm also trying to spread this around to cover a fairly representative set of subfields; having done classical mechanics and E&M at some length, I need to rush through a couple of other subfields quickly. One of these is classical thermodynamics, a field with a rich history and wide…
drorzel | December 15, 2011Moving along through our countdown to Newton's birthday, we have an equation that combines two other titans of British science:
This is the third of Maxwell's equations (named after the great Scottish physicist James Clerk Maxwell), but it originates with Michael Faraday, one of the greatest experimentalists of the day. Faraday was a fascinating guy, who came from humble origins-- he was an apprentice bookbinder who managed to get a job as Humphrey Davey's assistant-- to become hugely influential in both chemistry and physics. He also played an important role in science communication and…
drorzel | December 14, 2011As we march on toward Newton's birthday, we come to the second of Maxwell's famous equations, which is Gauss's Law applied to magnetic fields:
For once, this is pretty much as simple as it looks. The divergence of the magnetic field is zero, full stop. As I said yesterday (albeit using the wrong terminology), the left-hand side of this equation basically means that you look at the magnetic field in the vicinity of some point in space, and ask how many little arrows point toward the point of interest versus how many point away. What this equation tells us is that no matter where you look,…
drorzel | December 13, 2011As the advent calendar moves into the E&M portion of the season, there are a number of possible ways to approach this. I could go with fairly specific formulae for various aspects, but that would take a while and might close out some other areas of physics. In the end, all of classical E&M comes down to four equations, known as Maxwell's equations (though other people came up with most of them), so we'll do it that way, starting with this one:
This is the first of Maxwell's equations, written in differential form, and this relates the electric field E to the density of charge in…
drorzel | December 12, 2011Having covered most of what you need to know about classical physics, the traditional next step is to talk about electricity and magnetism, colloquially known as "E&M," though really, "E and B" would be more appropriate, as the fundamental quantities discussed are the electric field (symbol: E) and the magnetic field (symbol: B), whose effect is given by today's equation:
This is the "Lorentz force law," giving the force experienced by a particle with charge q moving at a velocity v through a region with both electric and magnetic fields. This is, in some sense, what defines those…
drorzel | December 12, 2011We kicked off our countdown to Newton's birthday with his equations of motion, so it seems fitting to close out the section on classical mechanics with another of Newton's equations, this time the Law of Universal Gravitation:
Like all the other equations to this point, I'm cribbing this from the formula sheet for my just-completed intro mechanics class, which means it's in the notation used by Matter and Interactions. This is sub-optimal in some ways-- I prefer to have subscripts on the r to remind you which way it points, but I don't care enough to re-do the equation.
So, this is the…
drorzel | December 10, 2011Today's advent calendar post was delayed by severe online retail issues last night and child care today, but I didn't want to let the day pass completely without physics, so here's the next equation in our countdown to Newton's birthday:
This is the final piece of the story of angular momentum, the undefined symbol from the right-hand side of the angular momentum principle: torque is defined as the cross-product between the radius vector pointing out from the axis of rotation to the point where the force is applied, and the vector force that acts at that point. As with the definition of…
drorzel | December 9, 2011Now that we've defined angular momentum, the next equation on our countdown to Newton's birthday tells us what to do with it:
This is the Angular Momentum Principle, and as with energy and momentum before it, this relates the time derivative of the angular momentum (that is, how quickly it's changing its value) to a quantity related to the interactions with other objects, in this case the torque.
So, why is this important?
As with energy, with the proper choice of system, we can often ensure that there is no net external torque on the system, in which case the right-hand side of this…
drorzel | December 8, 2011Moving along through our countdown to Newton's birthday, we come to the next important physical quantity, angular momentum. For some obscure reason, this gets the symbol L, and the angular momentum for a single particle about some point A is given by:
This is probably the most deceptive equation we'll see this season. Yesterday's definition of work clearly showed its vector calculus roots, but to the untrained eye, this just looks like a simple multiplication: You take the momentum (p) and multiply by the distance (r) from point A, and you're all set.
To those with a little mathematical…
drorzel | December 7, 2011Following the basic pattern established at the start of our seasonal countdown to Newton's birthday, today's equation defines a piece that was left hanging in yesterday's post:
This is the technical definition of "work" in physics terms. It's also probably the scariest-looking equation to this point, as it explicitly involves vector calculus-- there's an integral sign, and a dot product. The basic concept is simple enough, though: you look at the force F exerted on an object, multiply it by the distance dr that the object moves under the influence of that force, and then add up the Fdr…
drorzel | December 6, 2011For the sixth day of our advent countdown to Newton's birthday, we have the first equation that really departs from the usual notation. I've gotten to kind of like the way the Matter and Interactions curriculum handles this, though, so we'll use their notation:
This is what Chabay and Sherwood refer to as the Energy Principle, which is one of the three central principles of mechanics. The term on the left, ΔE represents the change in the total energy of a system, while the two terms on the right represent the work done on that system by its surroundings, and any heat energy flow into or out…
drorzel | December 5, 2011Moving along in our countdown to Newton's birthday, we start to deal with equations that Sir Isaac never would've seen, because they deal with more abstract quantities than he worked with. The first and in some ways most important of these is energy:
This is the full and correct expression for the energy of a particle with mass m moving at speed v. The notion of energy traces back to Newton's contemporary and rival Gottfried Wilhelm Leibniz, but this particular equation involves the same square-root factor as Saturday's definition of momentum. That tells you for sure that this particular…
drorzel | December 4, 2011Continuing our countdown to Newton's birthday, let's acknowledge the contributions of one of his contemporaries and rivals with today's equation:
This is, of course, Hooke's Law for a spring, which he famously published in 1660:
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Clears everything right up, doesn't it? OK, maybe not. This one's not only in Latin, it's a cryptogram, unscrambling to "ut tensio sic vis," which translates roughly to "as the extension, so the force," giving the correct proportionality between the force exerted by a spring or other elastic material and the amount that material has been stretched.…