A Recreational Physicist : Physics

Finaly something about the mathematics of physics.

andy — Thu, 13 Oct 2005 10:01:00 GMT

My biggest bane in college was the library, it sucked in most of my free time. One of my favorite things to do was browse the mathematics and physics stacks looking for interesting books. It was rare when I didn’t have a stack of 10 or 20 library books sitting on my bedroom floor. That’s also why I graduated with $300 in library fines and I didn’t get my diplomas until years later when I finally paid them off.

One of my favorite discoveries in the PMA library at UT was I. M. Yaglom, a Russian mathematician who authored a series of books aimed at advanced high school students or beginning college students focusing mostly on geometry. In my next few posts I plan to cover some of the topics from some of his books. They combine the algebra of complex numbers, the geometry of special relativity and string vibrations.

What I find even more interesting is that it provides a mathematical model for motion on a string which has some strong parallels with the free particle one dimensional Schroedinger equation.

To begin, I’d like to start with some basic properties of numbers and geometry. This is based on the Foundations of Geometry by David Hilbert. We first learn about numbers by counting. If we have an apple in our left hand and an apple in our right hand, then we have two apples. From counting we advance to addition and subtraction. This is modelled by adding and removing items from a group. If we have a basket with 10 apples and we eat 2 then we have 8 apples left.

Moving on we learn how to add groups of items. If we have 3 baskets containing 5 apples each, then we have 3 times 5 or 15 apples total. Multiplication is just addition of groups of smaller items. This model works fine for the positive model of addition and multiplication, but begins to break down for subtraction or division. Attempting to create a logical model of subtraction leads to the concept of negative numbers which don’t make much sense in terms of apples. Nobody has ever seen a negative apple. Reversing multiplication leads to division and the introduction of fractions which might or might not make sense for the objects we are dealing with.

In Foundations of Geometry Hilbert describes the construction of an algebra of line segments based on geometric operations between lines and line segments. In this view addition is simple, just join two line segments together and line them up. Subtraction is just as natural, all you have to do is reverse the direction of one of the segments. A negative number is just a line going in the opposite direction.

Multiplication is more complicated and I’ll cover that in the next post. In fact there are a couple of ways to define it, and these have to deal the two multiplication operations of vector algebra. If you don’t already know, think of how you would multiply two line segments together with just a ruler and compass. It sounds much easier then it is.

Hydrino state

andy — Sat, 06 Aug 2005 04:46:00 GMT

Medical doctor Randy Mills is claiming that there is a Hydrino state of hydrogen with a ground state energy lower then the regular ground state of hydrogen. From that he argues that you can get extra energy out of hydrogen atoms. This is a common pseudo-scientific claim for various free energy schemes.

Beyond being sceptical just because it hasn’t been discovered in around 100 years of electrical and collision experiments, there are a few other reasons not to trust these claims. First, even though Bohr’s theory of the hydrogen spectra was wrong, It is expressible in a simple quantization of the primary symmetry of the system, the angular momentum. This is something that holds true in the Bohr model, and the Schroedinger model. There is a powerful symmetry and correctness in the simple description this supplies. Even though it is not technically correct, and Quantum Electrodynamics is needed to compute energy levels that agree with the latest experiment. It comes very close to the true value because of the strength of the underling symmetry of hydrogen. These types of extraordinary claims need extra-ordinary levels of proof before they should even be considered.

the equations used to find the states are wrong and they are wrong in a specific way that author knows allows these states to exist

I’ve looked over the papers both positive and negative for Mills’ theory. The negative paper completely debunks the mathematical framework Mills constructed to support his theory. It doesn’t agree with quantum mechanics in many significant ways, including any other atom, or even higher energies of the hydrogen atom. The non-debunking paper looks at solutions to the hydrogen system from relativistic quantum mechanics. It starts by deriving the standard solutions of these equations and and then looks at solutions that are considered non-physical because they are part of a family of solutions which diverge at the center of the atom. The author then claims we should consider these solutions because we know the equations used to find the states are wrong and they are wrong in a specific way that author knows allows these states to exist, when there isn’t a lot of evidence for them in the first place.

No attempt is taken to anaylize non-spherical states or even higher energy atoms. If this theory is true, there are many other non-physical states that should be accessible and we should see them many more places then just in hydrogen plasmas. For each class of quantum system that physicists study this should add a near infinite number of states accessible and physical in this same way.

As far as free energy goes. Let’s suppose there are energy levels lower then the ground state energy of hydrogen and we can access them. How does this give us free energy? To get at this energy we need some sort of catalyst that will take regular hydrogen and convert it into the Hydrino state. In order to get energy from this, we have to convert a mass of hydrogen to this new state without it being able to convert back. Apparently once hydrogen is in this state it is only accessible through the collision with a catalyst atom.

This means that there are states of the electron which are invisible to the electro-magnetic force. The same force responsible for transferring energy in collisions. If it were able to interact with the electro-magnetic force then it would be able to exchange a photon with the local field and would be the standard ground state of hydrogen. Hmmmm. I can only see one problem with this, the electron is bound to the proton through the electro-magnetic force. So without even using mathematics and bringing equations into play there are enough holes in this theory to be highly suspicious. I give this theory a 9 out of 10 in the bogosity scale. Cold fusion is more likely then this.

Projective Geometry with Clifford Algebra

andy — Mon, 01 Aug 2005 01:30:00 GMT

A really nice application of Clifford Algebra to projective geometry. Check it out here.

Superlens

andy — Sat, 23 Apr 2005 16:59:00 GMT

I saw this article on PhysicsWeb. Negative index of refraction materials that operated with visible light were just announced a few weeks back, and already there is a lens that can focus images smaller then the wavelength of light. The really amazing thing is that these materials are made just by building a nano structure with standard materials rather then requiring any new exotic materials.

Superlens