Sunday, April 08, 2007

Elegance inscribed in the law of nature

Recently, I came across an article that describes how our nature seems to exhibit sheer beauty in the elegance of some physics equations and I can't help but to recall the one of the Psalms: "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they display knowledge." (Ps 19:1-2) The article which I am including below traces the development of an equation in modern physics and how it leads one to be amazed at God's creation if we presuppose that the world is created by Him. Of course, an atheist or agnostic with different presupposition may be inclined to think that it is all co-incidence and using the framework by David Hume, he would say that all these are orders imposed by our minds on the nature. But that would be another issue that would be addressed in another posting. In the meanwhile, may you enjoy reading the article below.


The remarkable advances made in spectroscopy (the branch of science concerned with the investigation and measurement of spectra produced when matter interacts with or emits electromagnetic radiation) during the nineteenth century allowed the spectrum of the sun to he examined for the first time, with results which proved to be of foundational importance. The presence of a hitherto unknown yellow line in that spectrum, without any known terrestrial parallel, led to the discovery of helium. Over the period 1859–60, the frequencies of a series of four lines observed in the solar spectrum were measured with what, for those days, was amazing accuracy – roughly one part in ten thousand (Angstrom 1868). The precision of these measurements of the visible line spectrum of atomic hydrogen led to the development of a new science: "spectral numerology" (Pais 1991, 142). This was an attempt to account for the relationship of the observed spectral lines with some fundamental mathematical equation.

The breakthrough, when it came, was simple and elegant. Working only on the basis of the four frequencies reported by Angstrom, J. J. Balmer found that he could exactly reproduce the frequencies by means of the following formula:
v = R (1/b2 – 1/a2)
where R is a constant now known as the "Rydberg constant" (3.29163 x 1015), b = 2, and a = 3, 4, 5 and 6 respectively. It must be stressed that this was an exact fit, not an approximation! Balmer (who was then a teacher in a Basle high school) mentioned his observations to the professor of physics at the University of Basle. By this time (1885), 12 more frequencies had been established, although this was unknown to Balmer. On learning of them from his colleague, Balmer found that they could all be fitted into his equation – again, exactly – without any difficulty, by setting a = 2 and b = 5, 6, ... 15, and 16. In fact, Balmer's formula allows an entire series of spectral lines to be predicted (one of which is now known by his name), as follows:
b = 1, a = 2, 3 . . . Lyman series (ultraviolet)
b = 2, a = 3, 4 ... Balmer series (visible)
b = 3, a = 4, 5 ... Paschen series (infrared)
b = 4, a = 5, 6 . . . Brackett series (far infrared)
b = 5, a = 6, 7 . . . Pfund series (far infrared)
b = 6, a = 7, 8 ... Humphreys series (far infrared)

The sheer beauty of the equation – matched, it has to be said, by experimental evidence, both retrodictive (i.e., accounting for something which was already known) and predictive....

By March 6, 1913, the Danish physicist Niels Bohr realized the significance of what Balmer had uncovered (Pais 1991, 143–55). On the basis of a quantum mechanical interpretation of the hydrogen atom, Bohr was able to derive Balmer's formula in two manners. For the first time, it became clear that Balmer's formula corresponded to aspects of the fundamental structure of the hydrogen atom.

The discovery opened the way to rapid development. Once more, the concept of "beauty" played an important part. During the period 1925–6, Werner Heisenberg and Erwin Schrodinger were both working on ways of describing atomic events (Pais 1991, 267–89), especially in the light of the work of Louis de Broglie. Their younger colleague Paul Dirac describes their different approaches as follows (Dirac 1963, 46–7):

Heisenberg worked keeping close to the experimental evidence about spectra .. . Schrodinger worked from a more mathematical point of view, trying to find a beautiful theory for describing atomic events . . . He was able to extend de Broglie's ideas and to get a very beautiful equation, known as Schrodinger's wave equation, for describing atomic processes. Schrodinger got this equation by pure thought, looking for some beautiful generalization of de Broglie's ideas, and not by keeping close to the experimental development of the subject in the way Heisenberg did.

The differences in approach are highly significant. Heisenberg worked outwards from the experimental evidence; Schrodinger sought an elegant theory which would then account for that evidence. The two, as it proved, converged. The quest for beauty and the quest for truth met at a common point. This point is clearly hinted at in Heisenberg's reflections on his work (Heisenberg 1971, 59, 68):

I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me . . . If nature leads to mathematical forms of great simplicity and beauty – coherent systems of hypotheses, axioms, etc...we cannot help thinking that they are "true," that they reveal genuine features of beauty.

The general drift of this analysis will be clear. A strong doctrine of creation (such as that associated with Christianity) leads to the expectation of a fundamental convergence of truth and beauty in the investigation and explanation of the world, precisely on account of the grounding of that world in the nature of God. The correlation in question is not arbitrary or accidental, but corresponds to the reflection of the nature of the creator in the ordering and regularity of creation.

Extracted from:
Alister E. McGrath, The Foundations of Dialogue in Science & Religion, Blackwell, 1998, pp. 77-79

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