A harmonic function is defined as one that satisfies Laplace's equation,
For cylindrical symmetry, the Laplacian (the operator represented by the top-heavy triangle squared) takes the following form:
This is where a neat trick is used. We make an assumption that the amplitude of the wave, denoted here by ψ, can be represented as a product of three separate functions which each only depend on one coordinate. To be more explicit,
This technique is known as "separation of variables." We claim that the function, ψ, can be separated into a product of functions each with their own unique variable. The results of this mathematical magic are astounding, since it greatly simplifies the problem at hand. When you go through the rigamarole of plugging this separated function back in, you get three simpler equations, each with its own variable.
Notice that the partial derivatives have become total derivatives, since these functions only depend on one variable. These are well-known differential equations in the mathematical world; the Φ function is a linear combination of sin(nϕ) and cos(nϕ) (this azimuthal angle, ϕ, goes from 0 to 2π and cycles, so this isn't terribly surprising) with n being an integer, and the Z function is a linear combination of cosh(kz) and sinh(kz), which are the hyperbolic functions. These equations are not what I want to focus on; what we've really been working so hard to get is the radial equation:
This is Bessel's differential equation. The solutions to this equation are transcendental (meaning that you can't write them as a finite sum of polynomials; the sine and cosine functions are also transcendental). We write them as
The Jn are finite at the origin (J0 is 1 at the origin, all other Jn are 0), and the Yn are singular (undefined) at the origin. They look something like this:
The Jn are much more common to work with because they don't have infinities going on, but the Yn are used when the origin is inaccessible (like a drum head that has a hole cut in the middle). These harmonic functions are used to model (but are not limited to)
- Vibrational resonances of a circular drum head
- Radial wave functions for potentials with cylindrical symmetry in quantum mechanics
- Heat conduction in a cylindrical object
- Light traveling in a cylindrical waveguide
There are some cool videos (this one has a strobe effect during it) showing them in action. There are also some cool Mathematica Demonstrations related to them as well. There are also orthogonality relationships with them, but I'll save that for another day.
Happy birthday, Tekky. I hope you have some part of the weekend off to make up for your busy day today.
P.S. Go over here to wish Tekulo a happy birthday yourself! 8D
So, I'm now at home doing work since there's actually power here. So that's been fun.
Outside looks especially pretty, though.
They're graphs in the complex plane. The color indicates the phase, or argument, of the complex number, and for this function, curves of equal phase are hyperbolas. To animate it, all I did was let the phase vary linearly in time.
"…You can always tell the particles apart, in principle—just paint one of them red and the other one blue, or stamp identification numbers on them, or hire private detectives to follow them around."
"...And, of course, if you’re in a really bad mood you can create a state for which neither position nor momentum is well defined..."
"It is traditional to write the Bohr radius with a subscript: a0. But this is cumbersome and unnecessary, so I prefer to leave the subscript off."
"If you think this is starting to sound like a mystical numerology, I don’t blame you. We will not be using Clebsch-Gordan tables much in the rest of the book, but I wanted you to know where they fit into the scheme of things, in case you encounter them later on. In a mathematical sense this is all applied group theory—what we are talking about is the decomposition of the direct product of two irreducible representations of the rotation group into a direct sum of irreducible representation (you can quote that, to impress your friends)."
"I’m not at all sure what I’m supposed to say today. Maybe you’re expecting a grand philosophy of education. But I learned very early as a parent that almost any philosophy of childrearing is worse than no philosophy at all, and I am inclined to think the same applies to teaching."
"Personally, I never bring notes to a lecture unless I am egregiously ill-prepared, for they break a very delicate and important bond of trust with the listener: If B really follows from A, how come he has to refer to his notes?"
"There are a thousand ways to get a problem wrong—not all of them bad—and many ways to get a problem right—not all of them good."
"Above all, I think studying science—and especially physics—is a tremendously liberating experience. I don’t happen to know how a carburetor works; I’m not even sure what a carburetor does; let me be frank: I don’t know what a carburetor looks like. But I do know that the behavior of carburetors is perfectly rational; somebody understands them, and if I really wanted to I’m sure I could understand them too. For I have confidence, grounded in the study of physics, that the world is rationally intelligible, and this, to me, is the most important—and most profoundly liberating—idea in human experience. The universe is comprehensible..."
"A colleague of mine in Chemistry likes to boast that ‘‘anyone can teach; the important thing is to attract good researchers.’’ I think it’s exactly the reverse: competent research physicists are a dime a dozen, but good teachers are few and far between. Please don’t misunderstand: I’ve got nothing against research—I do a certain amount of it myself, and I think it goes hand in hand with good teaching. But I regard myself as a professional teacher, and an amateur researcher, whereas most physicists are professional researchers but amateur teachers, and it shows. In my opinion by far the most effective thing we can do to improve the quality of physics instruction—much more important than modifications in teaching technique—is to hire, honor, and promote good teachers."
There are many more wonderful quotes, but I don't remember them/don't have the sources on me. Perhaps I'll add to this in another blog entry.
- My friend was trying to talk to me about atoms, but I got Bohr'd.
- Did you hear that Albert Einstein developed a theory about space? It was about time, too.
- Never trust an atom; they make up everything.
- The oddly pleasant feeling of looking down on a physicist as they finish the last of their drink. The strange charm of a top-down bottoms-up.
- Why does hamburger have less energy than steak? It's in the ground state.
- Why are physics books always unhappy? Because they're full of problems.
- Neutrinos make the worst friends; they rarely interact with anyone.
- In a quantum finish!
- KK, Tekulo, and I are in the same house again. This changes tomorrow, as KK and I are going to a friend's house, then KK returns to school.
- Had a New Year's Eve celebration with friends from college. Had a blast.
- My Christmas gifts include Super Mario Bros. 3D World, the Winter Market LEGO set (which I may trade with KK for his Winter Village Cottage, which I like a lot), a Fluttershy (my third so far), the mini-VW camper LEGO set, and a ceramic Snoopy sitting on his doghouse. All in all, a good haul.
or, in component form,
The word "virial" comes from the Latin vis, which means "force" or "energy," and looking at the equation, it makes sense why it's called that. Here the big Σ means sum, the "k" index denotes the kth particle of a system of N particles, V is the potential energy function affecting the kth particle, T is the potential energy of all the particles in the system, and rk is the position of the kth particle. This essentially relates the kinetic energy of all the particles to the positions and forces exerted on each particle (since -grad V is the force when energy is conserved, which is an assumption we are making). The brackets 〈 〉 denote that we're taking an average, so 〈T〉 is the average kinetic energy, etc.
Now, you may be thinking, "okay, that's a cute equation, I guess, but I don't see how it's particularly useful." Okay, here's where the usefulness comes in. Let's say I want to know the mass of some distant galaxy, but I don't have a good galaxy-weighing device on hand. We know that the gravitational potential energy of an object is given by
where m is the mass of the star, M is the mass of the center of the galaxy, and r is the distance from the center of the galaxy. Taking the distance r and multiplying by the gradient of the potential yields...the potential again, with a negative sign out front. So, for gravity,
Plugging this into the Virial theorem above and noting that 2T = mv^2 (where v is speed), we get that, for an object in the gravitational pull of an object of mass M,
Thus, we have at our disposal a way of measuring the mass of something like a galaxy by measuring only the speeds of stars and their distance away from the center. That's pretty incredible.
This actually is one of the ways scientists support the idea that there is dark matter in the universe; the Virial theorem gives an average of what speeds the stars in our galaxy should have based on their distance away from the center of the Milky Way, but what we actually observe is startlingly different. Thus, we can conclude that something is wrong with our knowledge of how gravity within a galaxy works. Based on this and other observations, the idea that there's extra stuff that can't be seen that adds to the gravitational force of a galaxy seems to be a reasonable idea.
In my research on diatomic hydrogen (H2), the Virial theorem is used in a different capacity. When figuring out the potential energy of an electron (or two) around the two positively charged protons, the virial has the Coulomb force term (which is just -V, just like gravity) and an additional term that pops up from assuming that the electrons are keeping the protons at equilibrium. I won't go too much into the physics, but the final product is
where E, T, and V are the total energy, kinetic energy, and potential energy of the electron(s), respectively, and R is the distance between the nuclei. This tells us something useful about the energy of the electrons; more specifically, it tells us about how the energy changes as you move the nuclei farther apart or closer together. In other words, since E = T + V,
which is very useful when constructing potential energy curves for hydrogen.
On a slightly related note, our lab's paper got published! Akano is now a for reals, published scientist! 8D
Akano Toa of Electricity
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Real Name: Forever Shrouded in Mystery
Likes: Science, Math, LEGO, Bionicle, Ponies, Comics, Yellow, Voice Acting
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Misses the 90's. A lot.
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