Quantum Theory

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1. 6 SUMMER/FALL 2000 by CATHRYN CARSON HAT IS A QUANTUM THEORY? We have been asking that question for a long time, ever since Max Planck introduced the element of dis-…
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  • 1. 6 SUMMER/FALL 2000 by CATHRYN CARSON HAT IS A QUANTUM THEORY? We have been asking that question for a long time, ever since Max Planck introduced the element of dis- continuity we call the quantum a century ago. Since then, the chunkiness of Nature (or at least of our theories about it) has been built into our basic conception of the world. It has prompted a fundamental rethinking of physical theory. At the same time it has helped make sense of a whole range of pe- culiar behaviors manifested principally at microscopic levels. From its beginning, the new regime was symbolized by Planck’s constant h, introduced in his famous paper of 1900. Measuring the world’s departure from smooth, continuous behavior, h proved to be a very small number, but different from zero. Wherever it appeared, strange phenomena came with it. What it really meant was of course mysterious. While the quantum era was inaugurated in 1900, a quantum theory would take much longer to jell. Intro- ducing discontinuity was a tentative step, and only a first one. And even thereafter, the recasting of physical theory was hesitant and slow. Physicists pondered for years what a quantum theory might be. Wondering how to inte- grate it with the powerful apparatus of nineteenth-century physics, they also asked what relation it bore to existing, “classical” theories. For some the answers crystallized with quantum mechanics, the result of a quarter- century’s labor. Others held out for further rethinking. If the outcome was not to the satisfaction of all, still the quantum theory proved remarkably THE ORIGINS OF THE QUANTUM THEORY W
  • 2. BEAM LINE 7 successful, and the puzzlement along the way, despite its frustrations, can only be called extraordinarily productive. INTRODUCING h The story began inconspicuously enough on December 14, 1900. Max Planck was giving a talk to the German Physical Society on the continuous spec- trum of the frequencies of light emitted by an ideal heated body. Some two months earlier this 42-year-old theorist had presented a formula capturing some new experimental results. Now, with leisure to think and more time at his disposal, he sought to provide a physical justification for his formula. Planck pictured a piece of matter, idealizing it somewhat, as equivalent to a collection of oscillating electric charges. He then imagined distributing its energy in discrete chunks proportional to the frequencies of oscillation. The constant of proportionality he chose to call h; we would now write e = hf. The frequencies of oscillation determined the frequencies of the emitted light. A twisted chain of reasoning then reproduced Planck’s postulated formula, which now involved the same natural constant h. Two theorists, Niels Bohr and Max Planck, at the blackboard. (Courtesy Emilio Segre` Visual Archives, Margrethe Bohr Collection)
  • 3. 8 SUMMER/FALL 2000 Looking back on the event, we might expect revolutionary fanfare. But as so often in history, matters were more ambigu- ous. Planck did not call his energy elements quanta and was not inclined to stress their discreteness, which made little sense in any familiar terms. So the meaning of his procedure only gradually be- came apparent. Although the problem he was treating was pivotal in its day, its implications were at first thought to be confined. BLACKBODIES The behavior of light in its interaction with matter was indeed a key problem of nineteenth-century physics. Planck was interest- ed in the two theories that overlapped in this domain. The first was electrodynamics, the theory of electricity, magnetism, and light waves, brought to final form by James Clerk Maxwell in the 1870s. The second, dating from roughly the same period, was thermo- dynamics and statistical mechanics, governing transformations of energy and its behavior in time. A pressing question was whether these two grand theories could be fused into one, since they started from different fundamental notions. Beginning in the mid-1890s, Planck took up a seemingly narrow problem, the interaction of an oscillating charge with its elec- tromagnetic field. These studies, however, brought him into con- tact with a long tradition of work on the emission of light. Decades earlier it had been recognized that perfectly absorbing (“black”) bodies provided a standard for emission as well. Then over the years a small industry had grown up around the study of such objects (and their real-world substitutes, like soot). A small group of theorists occupied themselves with the thermodynamics of radiation, while a host of experimenters labored over heated bod- ies to fix temperature, determine intensity, and characterize deviations from blackbody ideality (see the graph above). After sci- entists pushed the practical realization of an old idea—that a closed tube with a small hole constituted a near-ideal blackbody—this “cavity radiation” allowed ever more reliable measurements. (See illustration at left.) An ideal blackbody spectrum (Schwarzer Körper) and its real-world approximation (quartz). (From Clemens Schaefer, Einführung in die Theoretische Physik, 1932). Experimental setup for measuring black- body radiation. (The blackbody is the tube labeled C.) This design was a prod- uct of Germany’s Imperial Institute of Physics and Technology in Berlin, where studies of blackbodies were pursued with an eye toward industrial standards of luminous intensity. (From Müller- Pouillets Lehrbuch der Physik, 1929).
  • 4. BEAM LINE 9 Now Planck’s oscillating charges emitted and absorbed radiation, so they could be used to model a blackbody. Thus everything seemed to fall into place in 1899 when he reproduced a formula that a col- league had derived by less secure means. That was convenient; every- one agreed that Willy Wien’s formula matched the observations. The trouble was that immediately afterwards, experimenters began find- ing deviations. At low frequencies, Wien’s expression became increasingly untenable, while elsewhere it continued to work well enough. Informed of the results in the fall of 1900, on short notice Planck came up with a reasonable interpolation. With its adjustable constants his formula seemed to fit the experiments (see graph at right). Now the question became: Where might it come from? What was its physical meaning? As we saw, Planck managed to produce a derivation. To get the right statistical results, however, he had to act as though the energy involved were divided up into elements e = hf. The derivation was a success and splendidly reproduced the experimental data. Its mean- ing was less clear. After all, Maxwell’s theory already gave a beau- tiful account of light—and treated it as a wave traveling in a con- tinuous medium. Planck did not take the constant h to indicate a physical discontinuity, a real atomicity of energy in a substantive sense. None of his colleagues made much of this possibility, either, until Albert Einstein took it up five years later. MAKING LIGHT QUANTA REAL Of Einstein’s three great papers of 1905, the one “On a Heuristic Point of View Concerning the Production and Transformation of Light” was the piece that the 26-year-old patent clerk labeled revolutionary. It was peculiar, he noted, that the electromagnetic theory of light assumed a continuum, while current accounts of matter started from discrete atoms. Could discontinuity be productive for light as well? However indispensable Maxwell’s equations might seem, for some interesting phenomena they proved inadequate. A key example was blackbody radiation, which Einstein now looked at in a way dif- ferent from Planck. Here a rigorously classical treatment, he showed, Experimental and theoretical results on the blackbody spectrum. The data points are experimental values; Planck’s formula is the solid line. (Reprinted from H. Rubens and F. Kurlbaum, Annalen der Physik, 1901.)
  • 5. 10 SUMMER/FALL 2000 yielded a result not only wrong but also absurd. Even where Wien’s law was approximately right (and Planck’s modification unnecessary), elementary thermodynamics forced light to behave as though it were localized in discrete chunks. Radiation had to be parcelled into what Einstein called “energy quanta.” Today we would write E = hf. Discontinuity was thus endemic to the electromagnetic world. Interestingly, Einstein did not refer to Planck’s constant h, believing his approach to be different in spirit. Where Planck had looked at oscillating charges, Einstein applied thermodynamics to the light itself. It was only later that Einstein went back and showed how Planck’s work implied real quanta. In the meantime, he offered a fur- ther, radical extension. If light behaves on its own as though com- posed of such quanta, then perhaps it is also emitted and absorbed in that fashion. A few easy considerations then yielded a law for the photoelectric effect, in which light ejects electrons from the sur- face of a metal. Einstein provided not only a testable hypothesis but also a new way of measuring the constant h (see table on the next page). Today the photoelectric effect can be checked in a college labo- ratory. In 1905, however, it was far from trivial. So it would remain Pieter Zeeman, Albert Einstein, and Paul Ehrenfest (left to right) in Zeeman’s Amsterdam laboratory. (Courtesy Emilio Segre` Visual Archives, W. F. Meggers Collection)
  • 6. BEAM LINE 11 for more than a decade. Even after Robert Millikan confirmed Einstein’s prediction, he and others balked at the underlying quan- tum hypothesis. It still violated everything known about light’s wave- like behavior (notably, interference) and hardly seemed reconcil- able with Maxwell’s equations. When Einstein was awarded the Nobel Prize, he owed the honor largely to the photoelectric effect. But the citation specifically noted his discovery of the law, not the expla- nation that he proposed. The relation of the quantum to the wave theory of light would re- main a point of puzzlement. Over the next years Einstein would only sharpen the contradiction. As he showed, thermodynamics ineluctably required both classical waves and quantization. The two aspects were coupled: both were necessary, and at the same time. In the process, Einstein moved even closer to attributing to light a whole panoply of particulate properties. The particle-like quantum, later named the photon, would prove suggestive for explaining things like the scat- tering of X rays. For that 1923 discovery, Arthur Compton would win the Nobel Prize. But there we get ahead of the story. Before notions of wave-particle duality could be taken seriously, discontinuity had to demonstrate its worth elsewhere. BEYOND LIGHT As it turned out, the earliest welcome given to the new quantum concepts came in fields far removed from the troubled theories of radiation. The first of these domains, though hardly the most obvi- ous, was the theory of specific heats. The specific heat of a substance determines how much of its energy changes when its temperature is raised. At low temperatures, solids display peculiar behavior. Here Einstein suspected—again we meet Einstein—that the deviance might be explicable on quantum grounds. So he reformulated Planck’s prob- lem to handle a lattice of independently vibrating atoms. From this highly simplistic model, he obtained quite reasonable predictions that involved the same quantity hf, now translated into the solid- state context. There things stood for another three years. It took the sudden attention of the physical chemist Walther Nernst to bring quantum (From W. W. Coblentz, Radiation, Determination of the Constants and Verification of the Laws in A Dictionary of Applied Physics, Vol. 4, Ed. Sir Richard Glazebrook, 1923)
  • 7. 12 SUMMER/FALL 2000 theories of specific heats to general significance. Feeling his way towards a new law of thermodynamics, Nernst not only bolstered Einstein’s ideas with experimental results, but also put them on the agenda for widespread discussion. It was no accident, and to a large degree Nernst’s doing, that the first Solvay Congress in 1911 dealt precisely with radiation theory and quanta (see photograph below). Einstein spoke on specific heats, offering additional com- ments on electromagnetic radiation. If the quantum was born in 1900, the Solvay meeting was, so to speak, its social debut. What only just began to show up in the Solvay colloquy was the other main realm in which discontinuity would prove its value. The technique of quantizing oscillations applied, of course, to line spec- tra as well. In contrast to the universality of blackbody radiation, the discrete lines of light emission and absorption varied immensely from one substance to the next. But the regularities evident even in the welter of the lines provided fertile matter for quantum conjectures. Molecular spectra turned into an all-important site of research during The first Solvay Congress in 1911 assembled the pioneers of quantum theory. Seated (left to right): W. Nernst, M. Brillouin, E. Solvay, H. A. Lorentz, E. Warburg, J. Perrin, W. Wien, M. Curie, H. Poincaré. Standing (left to right): R. Goldschmidt, M. Planck, H. Rubens, A. Sommerfeld, F. Lindemann, M. de Broglie, M. Knudsen, F. Hasenöhrl, G. Hostelet, E. Herzen, J. Jeans, E. Rutherford, H. Kamerlingh Onnes, A. Einstein, P. Langevin. (From Cinquantenaire du Premier Conseil de Physique Solvay, 1911–1961).
  • 8. BEAM LINE 13 the quantum’s second decade. Slower to take off, but ultimately even more productive, was the quantization of motions within the atom itself. Since no one had much sense of the atom’s constitution, the venture into atomic spectra was allied to speculative model-building. Unsurprisingly, most of the guesses of the early 1910s turned out to be wrong. They nonetheless sounded out the possibilities. The orbital energy of electrons, their angular momentum (something like rotational inertia), or the frequency of their small oscillations about equilibrium: all these were fair game for quantization. The observed lines of the discrete spectrum could then be directly read off from the electrons’ motions. THE BOHR MODEL OF THE ATOM It might seem ironic that Niels Bohr initially had no interest in spec- tra. He came to atomic structure indirectly. Writing his doctoral thesis on the electron theory of metals, Bohr had become fascinated by its failures and instabilities. He thought they suggested a new type of stabilizing force, one fundamentally different from those famil- iar in classical physics. Suspecting the quantum was somehow implicated, he could not figure out how to work it into the theory. The intuition remained with him, however, as he transferred his postdoctoral attention from metals to Rutherford’s atom. When it got started, the nuclear atom (its dense positive center circled by electrons) was simply one of several models on offer. Bohr began working on it during downtime in Rutherford’s lab, thinking he could improve on its treatment of scattering. When he noticed that it ought to be unstable, however, his attention was captured for good. To stabilize the model by fiat, he set about imposing a quantum con- dition, according to the standard practice of the day. Only after a col- league directed his attention to spectra did he begin to think about their significance. The famous Balmer series of hydrogen was manifestly news to Bohr. (See illustration above.) He soon realized, however, that he could fit it to his model—if he changed his model a bit. He recon- ceptualized light emission as a transition between discontinuous or- bits, with the emitted frequency determined by DE = hf. To get the The line spectrum of hydrogen. (From G. Herzberg, Annalen der Physik, 1927)
  • 9. 14 SUMMER/FALL 2000 orbits’ energies right, Bohr had to introduce some rather ad hoc rules. These he eventually justified by quantization of angular momentum, which now came in units of Planck’s constant h. (He also used an in- teresting asymptotic argument that will resurface later.) Published in 1913, the resulting picture of the atom was rather odd. Not only did a quantum condition describe transitions between levels, but the “stationary states,” too, were fixed by nonclassical fiat. Electrons certainly revolved in orbits, but their frequency of rev- olution had nothing to do with the emitted light. Indeed, their os- cillations were postulated not to produce radiation. There was no predicting when they might jump between levels. And transitions generated frequencies according to a quantum relation, but Bohr proved hesitant to accept anything like a photon. The model, understandably, was not terribly persuasive—that is, until new experimental results began coming in on X rays, energy levels, and spectra. What really convinced the skeptics was a small modification Bohr made. Because the nucleus is not held fixed in space, its mass enters in a small way into the spectral frequencies. The calculations produced a prediction that fit to 3 parts in 100,000— pretty good, even for those days when so many numerical coinci- dences proved to be misleading. The wheel made its final turn when Einstein connected the Bohr atom back to blackbody radiation. His famous papers on radiative transitions, so important for the laser (see following article by Charles Townes), showed the link among Planck’s blackbody law, discrete energy levels, and quantized emission and absorption of radiation. Einstein further stressed that transitions could not be predicted in anything more than a probabilistic sense. It was in these same papers, by the way, that he formalized the notion of particle-like quanta. THE OLD QUANTUM THEORY What Bohr’s model provided, like Einstein’s account of specific heats, was a way to embed the quantum in a more general theory. In fact, the study of atomic structure would engender something plausibly called a quantum theory, one that began reaching towards a full-scale replacement for classical physics. The relation between the old and What Was Bohr Up To? BOHR’S PATH to his atomic model was highly indirect. The rules of the game, as played in 1911–1912, left some flexibility in what quantum condition one imposed. Initially Bohr applied it to multielectron states, whose allowed energies he now broke up into a discrete spectrum. More specifically, he picked out their orbital fre- quencies to quantize. This is exactly what he would later proscribe in the final version of his model. He rethought, however, in the face of the Balmer series and its simple numerical pattern fn = R (1/4 - 1/n2 ). Refocusing on one electron and highlighting excited states, he reconceptualized light emission as a transition. The Balmer series then resulted from a tumble from orbit n to orbit 2; the Rydberg constant R could be determined in terms of h. However, remnants of the earlier model still appeared in his paper.
  • 10. BEAM LINE 15 the new became a key issue. For some features of Bohr’s model preserved classical theories, while others presupposed their break- down. Was this consistent or not? And why did it work? By the late 1910s physicists had refined Bohr’s model, providing a relativistic treatment of the electrons and introducing additional “quantum numbers.” The simple quantization condition on angular momentum could be broadly generalized. Then the techniques of nineteenth-century celestial mechanics provided powerful theoret- ical tools. Pushed forward by ingenious experimenters, spectroscopic studies provided ever more and finer data, not simply on the basic line spectra, but on their modulation by electric and magnetic fields. And abetted by their elders, Bohr, Arnold Sommerfeld, and Max Born, a generation of youthful atomic theorists cut their teeth on such prob- lems. The pupils, including Hendrik Kramers, Wolfgang Pauli, Werner Heisenberg, and Pascual Jordan, practiced a tightrope kind of theo- rizing. Facing resistant experimental data, t
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