Carl’s new blog, “Ethanol Fuel”

During the past 24 hours, I heard, I swear!, 6 commentators blame biofuels for one of the world’s pains or another. What bothers me about this is that the complaints are, by and large, misguided in one way or another. So I’m starting a blog on the subject, “Ethanol Fuel”, or “Moses Lake”. It’s gotten to the point where the people in the industry joke with each other by pointing out yet another news article blaming biofuels for one thing or another.

So I’m starting another blog. I’m going to provide informed commentary on the energy situation, the food supply, land use, regulation, the economics of biofuels and fuels in general, the engineering, the politics of the subject, etc. And perhaps posting will leave me feeling that I’ve made a little effort towards stopping our country from being led by ignorance into policies that will take years to retract. I don’t think these belong on Mass which is devoted more to physics.

Producing fuel and electricity are engineering problems, the best decisions will be made by engineers who specialize in those areas, not by the general public. Nevertheless, the general public has the ultimate political power and their decisions, rational or otherwise, could effect the future of this country and this planet. And so I will write on the issues with the objective of education.

And it’s not like the US (or any other country) is immune to engineering decisions made on the basis of emotion. As a youth, I saw the nuclear power industry in the US destroyed by ignorance and fear masquerading as wisdom. Now, with the benefit of 30 years of hindsight, and another wave of ignorance and fear (this time related to global warming rather than radiation) support for nuclear power is increasing.

RIOFRIO Crushes CMB Anomalies!

Louise Riofrio recently pointed out that the inflation is in a bit of trouble due to the fact that it predicts a different curve than the one seen for the angular correlation of anisotropies of the Cosmic Microwave Background (CMB). An easily understood review of the CMB is given by The Cosmic Microwave Background for Pedestrians: A Review for Particle and Nuclear Physicists, astro-ph/0803.0834. The data excludes the curve expected by inflation at well above the 99% level:

Previously, Louise had explained the anomaly in a manner that I was too obtuse to understand, for example:

Views of the Cosmic Microwave Background may also indicate a spherical Universe. By measuring distances between acoustic peaks, scientists hope to complete a triangle and determine curvature. When a changing speed of light is accounted for, the angles do not add up to 180 degrees and the triangle is not flat. Most telling, the scale of density fluctuations is nearly zero for angles greater than 60 degrees. Like a ship disappearing over Earth’s horizon, the lack of large-angle fluctuations is smoking-gun evidence that the Universe is curved. Both lines of CMB data indicate that the curvature has radius R = ct.

David Hestenes’ electron model

John R Ramsden, as a comment on Motl’s blog, points us to an article by David Hestenes, Reading the Electron Clock, 0802.3227. There are two good reasons for looking at this paper, and the experiment that inspired it. First, we can use it as an excuse to discuss Hestenes’ electron theory, and what it looks like in the density operator language. Second, in a later post, we can discuss de Broglie’s matter waves.

Zitterbewegung

As the Wikipedia Zitterbewegung article states, Zitterbewegung [German for trembling motion] is an “interference between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median, with a circular frequency of , or approximately .” It was first noticed by Schoedinger, I think. It comes about when you compute the position operator, as a function of time, for an electron with momentum. The electron’s momentum p and mass m define a velocity v = p/m, and the position operator x ends up being given partly by vt = pt/m, but there is another term, an oscillatory term. [In quantum mechanics, the Hamiltonian H gives the energy, and setting H = mc^2, so the equation in the Wikipedia article looks like . Also, I'm mixing metaphors a bit between relativistic and non relativistic definitions of mass. Read the original articles for the correct derivation.] So long as you restrict to solutions of the Dirac equation which have only positive or negative energy, there is no Zitterbewegung, and this is how the frequency is usually ignored.

Hestenes is a long time advocate of The Zitterbewegung Interpretation of Quantum Mechanics. In his interpretation, Zitterbewegung (sometimes called zbw or zitter) is not due to interference between positive and negative frequencies but instead arises from the complex phase factor present in all quantum mechanics. Hestenes looks at the zbw as being associated with spin. The consequences are wide. He writes:

The essential feature of the zbw idea is the association of the spin with a local circulatory motion characterized by the phase factor. Since the complex phase factor is the main feature which the Dirac wave function shares with its nonrelativistic limit, it follows that the Schroedinger equation for an electron inherits a zbw interpretation from the Dirac theory. It follows that such familiar consequences of the Schroedinger theory as barrier penetration can be interpreted as manifestations of the zbw.

Lorentz Violation and Feynman’s Checkerboard Model

Lubos Motl brings to our attention a paper by Ted Jacobson and Aron C. Wall on black hole theremodynamics and Lorentz invariance, hep-ph/0804.2720 and claims that theories that violate Lorentz invariance are ruled out because they will also violate the second law of thermodynamics, the law that requires that entropy never decreases. Lubos concludes, “At any rate, this is another example showing that the “anything goes” approach does not apply to quantum gravity and if someone rapes some basic principles such as the Lorentz symmetry or any other law that is implied by string theory, she will likely end up not only with an uninteresting, ugly, and umotivated theory but with an inconsistent theory.” I disagree with this.

First, the abstract of the article:

Recent developments point to a breakdown in the generalized second law of thermodynamics for theories with Lorentz symmetry violation. It appears possible to construct a perpetual motion machine of the second kind in such theories, using a black hole to catalyze the conversion of heat to work. Here we describe the arguments leading to that conclusion. We suggest the implication that Lorentz symmetry should be viewed as an emergent property of the macroscopic world, required by the second law of black hole thermodynamics.

From the abstract, we see that Lubos has put the cart in front of the horse. Rather than proving that Lorentz symmetry has to be exact “all the way down”, the authors instead say that Lorentz symmetry does not have to be present at the foundations of elementary particles because it will automatically emerge macroscopically as a result of requiring that the second law of thermodynamics apply to black holes. And I agree wholeheartedly with this.

What is Intrinsic Quantum Spin?

In a sci.physics.foundations post, Jay Yablon has brought to light an obscure article by Hans C. Ohanian on the nature of the intrinsic spin of quantum objects and kindly loaded it onto the web: What is Spin? Am J. Phys. 54 (6) June 1986. The abstract is:

According to the prevailing belief, the spin of the electron or some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitivelyl appealing picture and establishes that neither the spin nor the magnetic moment are “internal” — they are not associated with the internal structure of the electron, but rather with the structure of the field. Furthermore, a comparison between calculations of angular momentum in the Dirac and electromagnetic fields shows that the spin of the electrons is entirely analogous to the angular momentum carried by a classical circularly polarized wave.

If you’re interested in the foundations of physics, the above is well worth reading. My efforts on quantum mechanics has been to look at things at a qubit level, where one reduces the number of degrees of freedom down to an absolute minimum. The calculations in the above are of momentum density, and energy density and the like. It’s nice to see them done explicitly.

Icosahedral Symmetry as a Decorative Knot

The decorative knots to be discussed here are those which are tied with one or more cords that may be repeated through several plies. These sorts of knots can be represented by self-intersecting loops on the plane, set up so that no more than two loops intersect at any single point. One generates a tying diagram from such by picking which of the two paths are uppermost at each intersection point. While this could be done more arbitrarily, for the knots discussed here the paths will be selected so that each path alternates over and under as in:

My eventual objective here is to tie a knot with approximate dodecahedral or icosahedral symmetry. Let’s begin with a line drawing that has the right symmetry. Flattened out to the plane, the dodecahedron looks like the following planar graph:

But this is not in the form we need; it is not in the form of a collection of loops. The basic problem is that, as a graph, there are three edges meeting at each vertex.

The Quantum Zeno Paradox or Effect

Kea recently brought up the subject of Zeno of Elea and his now long lost book of 40 paradoxes dealing with the continuum. His nominal 2500th birthday should be celebrated relatively soon. Let me paraphrase an example paradox is the following:

If one assumes that space and time are continuous, then an arrow shot from a bow, before reaching its target, must first travel half the distance. And then travel half the remaining distance. And so on. And therefore, there are an infinite number of distances to be travelled and the arrow could never reach the target. But arrows do reach targets. Therefore, space and time are not continuous.

Surprisingly, there is an echo of this thought in quantum mecahnics. The echo is so close to the original paradox that it is known as the Quantum Zeno’s Effect or sometimes “Paradox” depending on the writer. The subject is discussed in many arXiv articles.

In quantum mechanics, when one measures a system, the formalism requires that the system collapse to the result of the measurement. If one examines this carefully, one finds that if one measures a system at a sufficiently high rate, the effect of the repeated measurements is to prevent the quantum system from changing. In effect, if one examines the position of the arrow too frequently, the arrow cannot move. It’s worthwhile looking at the simple mathematics that causes this effect.

A new way to tie an old knot

The tying diagram Ashley gives for ABOK #2217 has a 4-fold axis of symmetry:

Tying a knot according to a diagram like this is quite time consuming. One must redraw the diagram by photocopying to the size needed. And in tying the knot, one pins the rope to the diagram. This is a pain because the rope moves around, the pins come out, etc. And the pins can damage the appearance of the rope.

In this post I give an alternative method of tying this knot, and several others like it, that is easier to set up, is much faster for each knot, and uses cheaper materials. Rather than an expensive cork board, we will use a 2×2 and build the knot as if it were a sort of Turk’s Head knot, on a cylindrical of square form.

Ashley knots; ABOK #2217 covering knot

For those of you who aren’t familiar with blue collar knots, “ABOK” means “The Ashley Book of Knots,” an ecyclopedic book on knots written in 1944 that has since become the reference for knot identification. I think it was my maternal grandfather that gave me my, somewhat rare, now the worse from love and use, 1st edition copy back in the 1960s; but what with the natural self-naturedness of a boy I cannot recall for sure. #2217 refers to a particularly handsome knot in the chapter “The Monkey’s Fist and Other Knot Coverings.” To justify “handsome” requires a sample photo from cbrew6 on Knot Heads World Wide:

This knot is “tied on the table,” which means that one uses a diagram to draw it. From a topological point of view, a table diagram in this case is a mapping of the surface of the sphere to the plane. A line drawing shows the path that the cord takes. The path is a loop, that is, it ends at the same point at which it starts. The path is restricted to never cross itself twice at a single point. At each crossing, some sort of notation indicates which line is to be on top, but for planar knots like the above, it is arranged so that the cord will weave over, under, over, under … And it is a trivial fact of practical folks topology that one can always assign such a pattern.

Quantum Bound States: the Hydrogen Atom

In quantum mechanics, bound states can be put together by choosing a potential energy and solving the a wave equation. The first non toy example that students learn is the hydrogen atom (or any other single electron atom) studied using Schroedginer’s wave equation. Our interest in this blog is more general bound state problems, but there is a lot we can learn by examining the hydrogen problem and its solutions.

The primary motivation for studying the hydrogen atom was to find an explanation for the light that was given off or absorbed when the atom switched from one energy state to another. For this reason, it was natural to look for the bound states as classified by their energies. That is, we will be looking for wave functions that correspond to sharp energies. If we wanted a more general solution, we can always combine different energy solutions by linear superposition.

The equation we wish to solve is where “H” is the quantum operator for energy (which is a sum of an operator for kinetic energy and one for potential energy), is the energy, is the wave function, and n =1, 2, 3, … is an index that distinguishes different energy solutions. It turns out the energy will be proportional to . Our solutions do not depend on time so we will leave off the t. And instead of writing (x,y,z), we will just write (x), or even leave it off completely like this: .