As the simplest of the stable atoms, hydrogen permits unique confrontation between theory and experiment.
—Theodor W. Hansch
The war changed I. I. Rabi's life and career. In his dual roles as the associate director of the MIT Radiation Laboratory and the director of research, Rabi worked closely with policy makers and military leaders. In the beginning, Rabi's discussions, particularly with military representatives, were infused with condescension, suspicion, and a benign refusal by generals and admirals to share vital military information. Rabi needed to know and understand the battle strategies being planned by the military in order to guide the development of radar equipment that would complement and support the war effort. But Rabi could play hardball himself, and soon his wisdom and intellectual toughness were recognized by Pentagon leaders. The unease dissipated, and frank, open discussions became the norm. This experience, coupled with his Nobel Prize in physics in 1944 and his close friendship with President Dwight D. Eisenhower, thrust Rabi into high levels of influence as the United States moved from World War II to the protracted Cold War.
Rabi's effectiveness during the war and the demands placed upon him in the years that followed meant that his focus changed and his productivity in his own Columbia research laboratory never reached the same level of intensity as it had before the war. As Rabi said, "The prize opens doors for you, doors which perhaps should not be opened. It attracts you away from your field; it brings many distractions. I'm not a man to open doors and tell people, "Do this, or do that." If I am invited in, however, then I can go into action. The Nobel Prize caused me to be invited, it brought the invitation. If I hadn't won the Nobel Prize, I wouldn't have had the temptation to respond to them . . . nor would I have had the obligation."1 Of course, Rabi could have made different choices, but he did not. Thus, Rabi's most significant postwar research results were those he reported in 1947 to the participants assembled at the Ram's Head Inn on Shelter Island.
The data Rabi presented on June 2, 1947 were based on results from a molecular beam experiment with hydrogen atoms. Before the war, Rabi's research on the hydrogens had focused on the measurement of the magnetic moments of the nuclei of the hydrogen atom: the proton and the deuteron. The experiment that Rabi and his two students, John Nafe and Edward Nelson, initiated when Rabi returned to Columbia University turned this prewar approach around. Because the magnetic moments of the proton and deuteron had been measured to high precision before the war by means of the magnetic resonance method, the new experiment took the values of the magnetic moments as a given and explored the energy states of the hydrogen atom themselves.
The two energy states that were the focus of Rabi's experiment were a direct consequence of magnetic influences at work within the hydrogen and deuterium atoms, arising from the magnetic moments of the electron, proton, and deuteron. The spectrum of hydrogen (and other atoms) originates from transitions between quantized energy states. However, in addition to energy, angular momentum is also quantized. Angular momentum arises from three sources: the orbital motion of the electron around the nucleus, the spin of the electron, and the spin of the nucleus. The quantized nature of angular momentum restricts the shapes of the orbitals allowed. The spins of both the electron and the nucleus give each an intrinsic magnetic moment. The electron and the nucleus can orient themselves only in certain ways relative to magnetic fields and can interact with each other. Each orientation corresponds to small differences in energy. The small splitting of the Bohr-type energy states because of relativistic effects and the electron's magnetic moment is called fine structure; the still smaller splitting arising from the influence of nuclear spins is called hyperfine structure. The complete energy level diagram of the energy states of hydrogen is shown in Figure 16.1.
The focus of the Rabi-Nafe-Nelson experiment was a transition between two hyperfine states in the ground state (the lowest n = 1 energy state) of the hydrogen atom; specifically the F = 0 and F = 1 states shown at the bottom of the right-hand column in Figure 16.1. Knowing the magnetic moments of both the proton and deuteron and assuming, on the basis of Dirac theory, that the magnetic moment of the electron was known exactly, the frequency separation of these two states could be calculated with precision. The calculated values were as follows:
frequency separation, hydrogen = 1416.90 ± 0.54 MHz and frequency separation, deuterium = 326.53 ± 0.16 MHz.
In their experiment, Rabi, Nafe, and Nelson used a beam of hydrogen atoms and measured the transition frequencies between the two hyperfine states. The measured values were as follows:
frequency separation, hydrogen = 1421.3 ± 0.2 MHz
and frequency separation, deuterium = 327.37 ± 0.03 MHz.
In many experiments, these results would be regarded as confirmation of theoretical expectations. After all, the discrepancy between the measured and calculated values was about 0.26 percent. However, these results illustrate the power and beauty of hydrogen: nothing is hidden. If theory and experiment do not agree exactly, something is wrong. For more complicated atoms, larger deviations between theory and experiment must be tolerated. The hydrogen atom is uncompromising.
The paper that reported these results ended with the recognition that there was a problem: "Whether the failure of theory and experiment to agree is because of some unknown factor in the theory of the hydrogen atom or simply an error in the estimate of one of the natural constants, such as [the fine structure constant], only further experiment can decide."2 This was the result that Rabi conveyed to the physicists at Shelter Island. Rabi's reputation as an experimentalist brought credibility to the measured results and issued a challenge to the theorists. As with the Lamb shift, it was quantum electrodynamics that was brought to bear on
Figure 16.1 The complete energy-level diagram of hydrogen showing fine and hyperfine structures. For an explanation of the first three columns, see the caption to Figure 15.1. The energy-state changes due to the interaction between the electron and nuclear magnetic moments are called hyperfine splittings. These splittings are smaller than the fine structure in the second column, but not as small as the splittings in Lamb shifts (third column). The two energy states in the lower right-hand corner are the two states on which the Rabi-Nafe-Nelson experiment focused and the source of the hydrogen 21-cm line. Not drawn to scale.
the small discrepancy and, in the process, QED exhibited its own potent power.
The culprit behind the disagreement between Rabi's experiment and Dirac's theory was the electron. Rabi's measurement was the separation of hyperfine energy states, which, as stated earlier, results from the interaction of the electron's magnetic moment with the proton's magnetic moment. Thus, the separation between these states was directly dependent on the magnitudes of both the electron's and proton's magnetic moments. Dirac theory had produced a value of the magnetic moment for the electron and its validity was, for all practical purposes, taken for granted. Such was the esteem for Dirac.3 Yet, in spite of its incredible success, Dirac theory was incapable of fully describing the hydrogen atom. The theory neither explained the Lamb shift nor provided an accurate description of the electron's magnetic moment.
It was quickly recognized that the explanation for the disparity between data and theory resided in the assumed value, Dirac's value, for the magnetic moment of the electron. "It's something I should have seen right off," said Rabi, "but I didn't."4 If the magnetic moment of the electron was slightly larger than that given by Dirac theory, harmony between experiment and theory could be restored. An experiment was designed by Polykarp Kusch, a colleague of Rabi's at Columbia, and Kusch's student Henry Foley to measure precisely the magnetic moment of the electron. The result was confirmed: the magnetic moment of the electron was slightly larger than predicted; hence, because it disagreed with Dirac theory, it was called anomalous. Given this experimental result, the challenge was to bring theory into agreement.
What is the electron? The electron, with both particle and wave properties, has four definite, quantitative properties: mass, charge, spin, and magnetic moment. Two of these properties, spin and the magnetic moment, seemed to be well accommodated by Dirac theory. (Why the electron has its particular charge and mass remains a mystery.) But there is more to the electron than meets the eye and it was quantum electrodynamics that revealed rather bizarre ways of conceptualizing the omnipresent electron.
So what is the electron? To answer this question, some insight into quantum electrodynamics is required. The principal actor in QED is the photon, which mediates the electromagnetic force. In the view of QED, the mechanism by which charges attract and repel each other is through the exchange of photons. As such, the electromagnetic field itself becomes quantized. The photon becomes the basic unit of the electromagnetic field. These photons have specific energies that are equal to hv where h is Planck's constant and v is the frequency of the photon. If atoms and photons exist together, they can interact with each other and atoms can absorb or emit photons.
A static charge, like an electron, takes on a new life in QED. An electron has an electromagnetic field consisting of quantized photons. Thus, the electron is surrounded by a cloud of photons. This cloud of photons surrounding an electron effectively reproduces the 1/r2 character of its measured electric field given by Coulomb's law. The electron can interact with its own electromagnetic field; that is, with photons in the cloud surrounding it. This interaction alters the behavior the electron would have in the absence of these interactions. To give a complete theoretical account of the electron interacting with its own field, corrections must be made by QED; in fact, by a new relativistic theory of QED. In the summer of 1947 Julian Schwinger recognized that to account for the results of the Rabi-Nafe-Nelson experiment, he would have to develop a relativistic QED, which he did with spectacular success during the next six months.
There is another consequence of the photon cloud around an electron. In this cloud of photons, the creation and annihilation of particles occur. It is these virtual particles, pairs of positive and negative particles, that lead to the polarization of the empty space surrounding the electron. Thus, the charge of the electron is partially screened from an outside viewer and, from a distance, appears slightly different from what it really is.
Incidentally, quantum electrodynamics transcends the electron. In other words, the ideas of QED go beyond the electron. For example, the concept of a basic interaction being mediated by an exchange of particles has been extended to both the weak and strong interactions with the mediating particles experimentally identified. The gravitational interaction is also assumed to be mediated by a particle, the graviton, but observing the graviton and quantizing the gravitational field has yet to be accomplished. The vacuum, again more generally, has taken on new life and it is viewed as teeming with particle creation and destruction.
The effect of the quantization of the electromagnetic field brings subtle changes to the energy states of the hydrogen atom as well as other atoms. These subtle changes are what Lamb measured. The QED corrections that were brought to bear on these changes accounted for the small shifts in the energy states.
So, given the character of QED, how did it bring an explanation to Rabi's data? Again, we ask: what is the electron? The electron does have a mass and a charge and, by virtue of the latter, the electron is the source of an electromagnetic field. The electron interacts with its own electromagnetic field and this interaction influences the mass and charge that is experimentally measured. The measured mass and charge are called the physical mass and physical charge. But what is the electron's mass and charge in the absence of these self-interactions? In the words of the trade, what are the bare mass and the bare charge? QED answered these questions and in the process revealed that the magnetic moment of the electron was slightly larger than that given by Dirac theory. It was this slightly larger magnetic moment of the electron that produced the unexpected deviation Rabi found in his data.
Stimulated by Rabi's data, experiments have been conducted to measure the magnetic moment of the electron with great precision. The precision achieved is astonishing. The electron's magnetic moment according to Dirac theory is
The most precise measurement of the electron's magnetic moment is slightly larger:
I = 1.0011596521884(43) Bohr magnetons where the (43) means there is an uncertainty in the last two figures of ± 43. Richard Feynman put the incredible precision of this measurement into perspective when he wrote, "This accuracy is equivalent to measuring the distance from Los Angeles to New York, a distance of over 3,000 miles, to within the width of a human hair."5
The agreement that exists between the measured value of the electron's magnetic moment and the theoretical value calculated with quantum electrodynamics is noteworthy. QED, the most successful theory in physics, is the standard by which other physical theories are judged. The value of the electron's magnetic moment calculated from QED rather than Dirac theory is
The agreement between the measured value of the electron's magnetic moment and the calculated value is 0.0001 part per million. As experimental methods become more and more refined and are capable of producing more and more accurate data, the world seems to get curiouser and curiouser. The natural world is much more imaginative than we ever imagined. When the imagination of humans held sway, human beings stood on a flat Earth positioned on the back of an elephant, which was supported on the back of a turtle, and so on. Instead, the Earth is a spinning ball hurtling around a nearby star and the human population stands on this globe—half of us upside-down relative to the other half. When human imagination was the guide, the matter of the universe consisted of earth, water, air, and fire. Instead, matter consists of atoms in motion, which in turn are made up of electrons, protons, and neutrons, and the proton and deuteron, in turn, consist of quarks and gluons. In addition, the so-called vacuum is pulsating with activity.
In arriving at insights into nature's bountiful imagination, we are fortunate that nature gave us the simple hydrogen atom. Its one electron with its nucleus of one proton or one deuteron has stimulated the feeble imaginations of scientists to probe behind the common-sense appearance of things and to soar to ever new heights of understanding. The concepts that have emerged from the laboratory have proven their power in synthesizing disparate realms of experience. At the same time, these concepts continue to challenge and boggle the best minds. As we look to the future, the hydrogen atom will continue to help us meet the challenge of embracing the natural world with understanding and, in the process, to understand better the place of humankind within the larger scheme of things.
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