On Bohr's Wave Theory of the Atom

2009 Mar 01

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    In a few years we will celebrate the first centennial of Niels Bohr's use of Planck's wave theory of the electron to produce the first accurate description of the atom. But we will celebrate more than Bohr. Late in life he compared himself to the upper stage of a rocket, saying that he had soared to such scientific heights because others had lifted him to a great height before he even started. I believe that we honor Bohr well by reviewing his boosters.

    Let's begin by looking at that great project of the Nineteenth Century, the effort to understand the nature of light. It began with William Herschel's discovery of infrared radiation when he conducted an experiment to measure the amounts of heat carried by the different colors in a ray of sunlight. It progressed through Gustav Robert Kirchhoff's challenge to his fellow physicists to devise a full mathematical description of the frequency distribution of heat in the radiation emanating from a perfectly black body. And we pick up the story in 1876.

    In that year Professor Adolfo Bartoli, at the University of Pavia, deduced the fact that light exerts pressure. True, Maxwell had deduced the same thing three years earlier, but until Heinrich Hertz began his experiments in generating and detecting electro-magnetic waves few physicists took Maxwell's work seriously enough to use it. Instead Bartoli exploited Rudolf Clausius's law of entropy: "No passive phenomenon can make net heat flow from a cooler to a warmer body."

    Inspired by Kirchhoff, Bartoli conceived a pulse of light as a kind of cloud-like body. That body would have the same temperature as the body that emitted it and the distribution of the frequencies of the radiation comprising the pulse would provide a measure of that temperature in accordance with Kirchhoff's law. If that pulse bounced off a mirror moving toward the pulse's source, then the pulse would come out of the collision with its frequencies increased uniformly by the Doppler shift, which would make the pulse a body of higher temperature in apparent violation of the law of entropy. Bartoli could only remove the violation by changing the collision from a passive process to an active one; he had to assert that the mirror did work upon the pulse sufficient to raise its heat content enough to keep its entropy unchanged. But that necessitates that the mirror exert a force upon the pulse, which necessitates in turn, via Newton's third law of motion, that the pulse exert a force upon the mirror. Thus Bartoli deduced that light exerts pressure upon any body that absorbs or reflects it.

    But in treating light in such a way, as if it has a gaseous nature, has consequences that other physicists exploited. Clausius saw in his imagination that light passing from one transparent medium to one with a different index of refraction would have to enact a phenomenon analogous to the Joule-Thomson throttling of a gas. When Clausius worked out a full description of the phenomenon he discovered a strange paradox. Part of his theory included a calculation of the energy density in the light and he obtained a result different from the energy density that we would calculate through Maxwell's electro-magnetic theory. Clausius found that he could only resolve the paradox by producing what the popular press calls the cottage-cheese theory of light, the statement that light consists of curds dispersed in an electro-magnetic whey. Each curd in that model can only carry one value of energy, a quantum equal to Clausius' constant multiplied by the frequency of the radiation.

    That odd idea inspired Ludwig Boltzmann to create his theory of blackbody radiation in 1895, which theory describes with complete accuracy the distribution of energy in the radiation emanating from a perfectly black body at a given temperature, thereby validating Clausius' strange result. Publication of that theory achieved the original goal that Kirchhoff had set, but Clausius' discovery reset the goal by showing that light has aspects to its nature that earlier physicists had not suspected.

    It was Max Planck who gave us the full wave-particle duality theorem in 1905 when he revisited his theory of the photoelectric effect from 1900. Using Einstein's mass-energy equivalence from the Theory of Invariants, Planck asserted that the wave-particle duality in the light striking a metal surface must be matched by a wave-particle duality in the electrons ejected from the metal. Returning the inspirational favor, Einstein then devised his famous wave equation, which he obtained by modifying the equation from classical mechanics relating Hamilton's function to the total energy in a system.

    Few physicists at that time took the new quantum theory seriously. But in 1913 Niels Bohr devised his theory of the atom. In that theory he applied Einstein's equation to a description of the hydrogen atom, assuming that the orbits of the electrons conform to self-reinforcing quantum waves. With the mathematical model he thus devised Bohr calculated the frequencies of radiation that a hydrogen atom would emit and found that they matched almost perfectly the frequencies measured in experimental observations of radiating hydrogen. In that way Bohr precipitated a phase change in physics, one that led physicists to engage in a massive collective effort to create and obtain experimental proof of the complete quantum theory of matter and radiation, the physics that is taught in undergraduate courses today.

    But consider what kind of world we might inhabit if something had gone wrong. What would our world look like if Clausius had ended up dismissing his paradox as a trivium irrelevant to the development of physics and had not published it? One of Professor Bohr's own students, Werner Heisenberg, noted in his history of the quantum revolution that the laws and equations that seem so familiar to us today would bear strange names in that alternate world. We might recall Shakespeare's aphorism that a rose by any other name would smell as sweet, but then Nöther's Indeterminacy Principle by another name would come later. Imagine a world in which the quantum theory did not appear until the 1920's or, even, the 1930's. Absent the invention of the transistor in 1939, the first digital computers would have used vacuum tubes. Such bulky, slow, and expensive machines would have evolved only slowly and we likely would not have seen the advent of the home computer in 1962. Absent the invention of the laser in 1945, optical computing would have come later, perhaps much later, than its actual advent in the 1980's.

    Would we have had the Manhattan Project in the early 1940's? Inspired by Einstein's letter to President Roosevelt, American scientists and engineers went to a place just east of the Town of Batavia, 32 miles west of Chicago, and built the mile-wide synchro-cyclotron, the Billion-Volt Big Boy, that has given us so much of our knowledge of the fundamental nature of matter. How little would we know now if our grandparents' generation had not built that big nuke-knocker?

    In deducing the full quantum theory of light and matter, Planck and Einstein revealed the power of human reason to find truths completely alien to human intuition. In using that theory to describe the atom in a way not otherwise possible, Niels Bohr presided over a very public wedding of scientific empiricism to a revived and renewed rationalism. Such a spectacular intellectual achievement could only have inspired a renewed Age of Reason. Out of the dark night of the Great War a new Enlightenment dawned over our world and the Romanticism of the Nineteenth Century faded away.

    What would our world look like if that had not happened and rationalism had remained an obscure intellectual sideshow in human affairs? Would the Romanticism have soured into a panoply of irrationalisms, such as the kind of tribalistic nationalism that led to the Great War? Could the League of Nations have survived and become the Grand Parliament of Humanity that we have today?

    James Booker has said that "there is none of them that knows how to appreciate a blessing until they are deprived of it." We thus appreciate the work of Niels Bohr and his predecessors and colleagues when we contemplate a world in which physics exists as a kind of pastiche of postulates and theorems rather than as the grand axiomatic-deductive structure that we have today.


    This little story is not a complete science-fictional joke. To see how Clausius actually could have deduced the quantum theory of light, go to "The Map of Physics", scroll down to the section titled "Radiation Thermodynamics", and read the essay titled "The Kirchoff-Clausius Law".


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