Quantum Physics World

Tuesday, March 28, 2006

One of the themes touched on in "Equations of Eternity" by David Darling is the unreasonable effectiveness of mathematics in describing the physical world. Time and again, as Darling points out, mathematicians have worked on some obscure theoretical idea or area that seems to have nothing to do with reality. Then years later physicists stumble on it and discover that it describes some physical process in absolute detail, down to the last dotted i and crossed t.
A great example of the connection between mathematics and the physical world is the discovery by Maxwell of well, Maxwell's equations. During the 19th century the frontiers of science were being pushed by people studying electromagnetic phenomena. Years earlier Coloumb had figured out how to describe the electric force between two charges. In the early to mid-1800's physics had moved quite a bit beyond that to consider electric currents and magnetic fields. It was here that mathematical insight would prove to be an unusually effective tool-revealing properties of nature hidden to the senses.
In about the 1830's Ampere worked out a "law" that relates the magnetic field to a flow of current. Ampere's law has a very precise mathematical form which was worked out from careful experimental observation. In short Ampere figured out that the curl of the magnetic field B is related to the current density J via



While this "law" was worked out by careful measurement and experimental observation-it violates a tenet of vector calculus. We know from vector calculus that the divergence of the curl of any vector is zero:






Maxwell basically applied this fact to Ampere's law and discovered that Ampere's law could not be true-based strictly on mathematical reasoning. This in turn led to the discovery of radio. Let's get a basic idea of what he did. And so, taking the divergence of the left hand side of Ampere's law, we should get zero



But on the right side, we have






But this isn't zero. In fact, using the conservation of current density together with Gauss's law, it can be shown that this is related to the electric field. That is






So, to make the divergence of Ampere's law vanish, Maxwell added this term. The missing term is the "displacement current" and it leads to a coupling of the electric and magnetic fields-giving us traveling electromagnetic waves or radiation. To sum up Maxwell changed Ampere's law to






These "laws" of vector calculus are abstract mathematical laws--supposedly laws of pure thought. At first sight one might not expect that they would hold precedence over experimental observation. But it turns out they do. Maxwell used the laws to determine what form Ampere's law should really have, and in the process discovered something that was unknown at the time-radio waves.
This is just one small example of the interplay between math and physics. Later we'll explore connections between abstract mathematics and quantum theory which describes every last detail of atomic behavior.

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