BOHR: ON THE WINGS OF A BUTTERFLY
Rutherford's discovery, coming after several experimental results that contradicted classical physics, was the last nail in the coffin. In the ongoing contest between experiment and theory, this would have been a good time to rub it in: "How clear do we experimenters have to make it before you theorists are convinced you need a new thing?" It appears that Rutherford didn't realize how much havoc his new atom was going to wreak on classical physics.
And then along came Niels Bohr, who would play Maxwell to Rutherford's Faraday, Kepler to his Brahe. Bohr's first position in England was at Cambridge, where he went to work for the great J. J., but the twenty-five-year-old kept irritating the master by finding mistakes in his book. While studying at the Cavendish Lab, on a Carlsberg Beer fellowship no less, Bohr heard Rutherford lecture about his new model of the atom in the fall of 1911. Bohr's thesis had been a study of "free" electrons in metals, and he was aware that all was not well with classical physics. He knew of course about Planck and about Einstein's more dramatic deviation from classical orthodoxy. And the spectral lines emitted by certain elements when they were heated was another clue to the quantum nature of the atom. Bohr was so impressed by Rutherford's lecture, and his atom, that he arranged to go to Manchester for a four-month visit in 1912.
Bohr saw the real significance in the new model. He realized that to satisfy Maxwell's equations, the electrons in circular orbits around a central nucleus would have to radiate energy, just like an electron accelerating up and down an antenna. To satisfy the laws of energy conservation, the orbits would shrink, and in no time flat the electron would spiral into the nucleus. If all these conditions were met, matter would be unstable. The model was a classical disaster! Yet there really was no alternative.
Bohr had no choice but to try something very new. The simplest atom of all is hydrogen. So Bohr studied the available data, such as how alpha particles slow down in hydrogen gas, and concluded that hydrogen has one electron in a Rutherford orbit around a positively charged nucleus. In facing up to a break with classical theory, Bohr was encouraged by other curiosities. He noted that nothing in classical physics determines the radius of the electron's orbit in the hydrogen atom. In fact, the solar system is a good example of a variety of planetary orbits. According to Newton's laws, any planetary orbit can be imagined; all it needs is to be started off properly. Once a radius is fixed, the planet's speed in orbit and its period (the year) are determined. But all hydrogen atoms, it would seem, are exactly alike. The atom shows none of the variety exhibited by the solar system. Bohr made the sensible but absolutely anticlassical assertion that only certain orbits are allowed in atoms.
Bohr also proposed that in these very special orbits the electron doesn't radiate. This, in historical context, was an incredibly audacious hypothesis. Maxwell rotated in his grave, but Bohr was simply trying to make sense of the facts. One important fact concerned the spectral lines that Kirchhoff had found shining out of atoms decades earlier. Glowing hydrogen, like other elements, emits a distinctive series of spectral lines. To get spectral lines, Bohr realized he must allow the electron to have the option of a number of different orbits corresponding to different energy contents. So he gave hydrogen's single electron a set of allowed radii representing a set of states of higher and higher energy. To explain spectral lines, he postulated (out of the blue) that radiation occurs when an electron "jumps" from one energy level to a lower one; the energy of the radiating photon is the difference of the two energy levels. He then proposed an absolutely outrageous rule for these special radii that determine the energy levels. Orbits are allowed, he said, in which the angular moment, a well-known quantity that measures the rotational momentum of the electron, takes on only integer values when measured in a new quantum unit. Bohr's quantum unit was nothing but Planck's constant, h. Bohr later said that "it was in the air to try to use the preexisting quantum ideas."
Now what is Bohr doing in his attic room late at night in Manchester with a sheaf of blank paper, a pencil, a sharp knife, slide rule, and some reference books? He is searching for nature's rules, rules that will correspond to the facts listed in his reference books. What right does he have to make up rules for the behavior of invisible electrons orbiting the nucleus (also invisible) of the hydrogen atom? His legitimacy is ultimately established by his success in explaining the data. He starts with the simplest atom, hydrogen. He understands that his rules ultimately have to emerge from some deep principle, but first the rules. This is how theorists work. Bohr in Manchester was, in the words of Einstein, trying to know the mind of God.
Bohr soon returned to Copenhagen to allow his seminal idea to germinate. Finally, in three papers published in April, June, and August of 1913 (the great trilogy), he presented his quantum theory of the hydrogen atom—a mixture of classical laws and totally arbitrary assertions (hypotheses) clearly designed to get the right answer. He manipulated his model of the atom so that it would explain the known spectral lines. Tables of these spectral lines, a series of numbers, had been painstakingly compiled by the followers of Kirchhoff and Bunsen, checked in Strasbourg and Göttingen, in London and Milan. What kind of numbers? Here are a few from hydrogen: λ1 = 4,100.4, λ2 = 4,339.0, λ3 = 4,858.5, λ4 = 6,560.6. (Sorry you asked? Don't worry. No need to memorize them.) How do these spectral vibrations come about? And why only these, no matter how the hydrogen is energized? Oddly, Bohr later minimized the importance of spectral lines: "One thought that spectra are marvelous. But it is not possible to make progress there. Just as if you had the wing of a butterfly, then certainly it is very regular with its colors and so on. But nobody thought that you could get the basis of biology from the coloring of the wing of a butterfly." And yet it turned out that the spectral lines of hydrogen, the wing of the butterfly, provided a crucial clue.
Bohr's theory was crafted to give the numbers for hydrogen that were on the books. Crucial to his analyses was the overriding concept of energy, a term that was precisely defined in Newton's time, then evolved and enlarged. An understanding of it is necessary for the educated person. So let's take two minutes for energy.
TWO MINUTES FOR ENERGY
In high school physics, an object with a certain mass and a certain velocity is said to have kinetic energy (energy by virtue of motion). Objects have energy also by virtue of where they are. A steel ball on top of the Sears Tower has potential energy because someone worked hard to get it up there. If you drop it off the tower, it will, in falling, trade in its potential energy for kinetic energy.
The only thing that makes energy interesting is that it is conserved. Picture a complex system of billions of atoms in a gas, all in rapid motion, colliding with the walls of the vessel and with one another. Some atoms may gain energy; others lose it. But the total energy never changes. It wasn't until the eighteenth century that scientists discovered that heat is a form of energy. Chemicals can release energy via reactions such as the burning of coal. Energy can and does continually change from one form to another. Today we recognize mechanical, thermal, chemical, electrical, and nuclear energy. We know that mass can be converted to energy via E = mc2. In spite of these complexities, we are still a hundred percent convinced that in complex reactions the total energy (including mass) always remains constant. Example: slide a block along a smooth plane. It stops. Its kinetic energy was changed into heat in the ever so slightly warmer plane. Example: you fill your car with gasoline, knowing that you have bought 12 gallons of chemical energy (measured in joules), which you can use to give your Toyota a certain kinetic energy. The gasoline goes away, but its energy can be accounted for—320 miles, from Newark to North Hero. The energy is conserved. Example: a waterfall crashes onto the rotor of an electric generator converting natural potential energy to electrical energy to warm and illuminate a distant town. In nature's ledger it all has to add up. You end up with what you brought.
SO?
Okay, how does this relate to the atom? In Bohr's picture, the elect
ron must confine itself to specific orbits, with each orbit defined by its radius. Each of the allowed radii corresponds to a well-defined energy state (or energy level) of the atom. The smallest radius corresponds to the lowest energy, which is called the ground state. If we pour energy into a sample of hydrogen gas, some of it is used in shaking up the atoms so that they move faster. Some of the energy, however; is absorbed by the electron in a very specific bundle (remember the photoelectric effect), which allows the electron to reach another of its energy levels, or radii. The levels are numbered 1, 2, 3, 4, ..., and each one has its energy, E1, E2, E3, E4, and so on. Bohr constructed his theory to include the Einstein idea that the energy of a photon determines its wavelength.
If photons of all wavelengths rain down on a hydrogen atom, the electron will eventually swallow the appropriate photon (light bundle of some particular energy) and jump up from E1 to E2 or E3, say. In this way electrons populate the higher energy levels of the atom. This is what happens, for example, in a discharge tube. When electrical energy goes in, the tube glows with the characteristic colors of hydrogen. The energy induces some electrons in the trillions of atoms to jump to higher energy states. If the input electrical energy is large enough, many of the atoms will have electrons occupying essentially all possible higher energy states.
In Bohr's picture the electrons in higher energy states spontaneously jump down to lower levels. Now remember our little lecture on the conservation of energy. If electrons jump down, they lose energy, and that lost energy has to be accounted for. Bohr said, "No problem." A dropping electron emits a photon of energy equal to the difference in energy of the orbits. If the electron jumps from level 4 to level 2, for example, the photon's energy is equal to E4 minus E2. There are lots of jump possibilities, such as E2→E1, E3→E1, or E4→E1. Multilevel jumps are also allowed, such as E4→E2, then E2→E1. Every change of energy results in the emission of a corresponding wavelength, and a series of spectral lines is observed.
Bohr's ad hoc, quasi-classical explanation of the atom was a virtuoso, if unorthodox, performance. He used Newton and Maxwell when they were convenient. He discounted them when they weren't. He used Planck and Einstein when they worked. It was outrageous. But Bohr was smart, and he got the right answer.
Let's review. Thanks to the work of Fraunhofer and Kirchhoff back in the nineteenth century, we knew about spectral lines. We knew that atoms (and molecules) emit and absorb radiation at specific wavelengths and that each atom has its own characteristic pattern of wavelengths. Thanks to Planck, we knew that light is emitted in quanta. Thanks to Hertz and Einstein, we knew that it is also absorbed in quanta. Thanks to Thomson, we knew there are electrons. Thanks to Rutherford, we knew that the atom has a dense little nucleus, lots of void, and electrons scattered throughout. Thanks to my mother and father, I got to learn this stuff. Bohr put this data—and much more—together. The electrons are allowed only certain orbits, said Bohr. They absorb energy in quanta, which forces them to jump to higher orbits. When they drop back down to lower orbits, they emit photons, quanta of light. Scientists observe these quanta as specific wavelengths—the spectral lines peculiar to each element.
Bohr's theory, developed between 1913 and 1925, is now referred to as the "old quantum theory." Planck, Einstein, and Bohr had each taken a step to flout classical physics. All had firm experimental data that told them they were right. Planck's theory beautifully agreed with the black body spectrum, Einstein's with detailed measurements of photoelectrons. In Bohr's mathematical formula one finds such quantities as the electron's charge and mass, Planck's constant, some it's, numbers like 3, and an important integer (the quantum number) that enumerated the energy states. All of these, when factored together; provided a formula from which all the spectral lines of hydrogen could be calculated. It was in remarkable agreement with the data.
Rutherford loved Bohr's theory but raised the question of when and how the electron decides to jump from one state to another—something Bohr didn't discuss. Rutherford remembered a previous puzzle: when does a radioactive atom decide to decay? In classical physics, every action has a cause. In the atomic domain that kind of causality doesn't seem to appear. Bohr recognized the crisis (which wasn't really solved until Einstein's 1916 work on "spontaneous transitions") and pointed out a direction. But the experimenters, still exploring the phenomena of the atomic world, found a number of things that Bohr hadn't counted on.
When the American physicist Albert Michelson, a precision fanatic, examined the spectral lines more closely he noticed that each of the hydrogen lines was actually two narrowly spaced lines—two wavelengths that were very close together. This doubling of lines means that when an electron is ready to jump down, it has a choice of two lower energy states. Bohr's model didn't predict the doubling, which was called "fine structure." Arnold Sommerfeld, a contemporary and associate of Bohr, noticed that the velocity of electrons in the hydrogen atom is a significant fraction of the velocity of light and should be treated in accordance with Einstein's 1905 theory of relativity. He made the first step toward joining the two revolutions, quantum theory and relativity. When he included the effects of relativity, he noted that where the Bohr theory predicted one orbit, the new theory predicted two closely spaced orbits. This explained the doubling of the lines. In carrying out this calculation, Sommerfeld introduced a "new abbreviation" of some constants that frequently appeared in his equations. It was 2πe2/hc, which he abbreviated with the Greek letter alpha (α). Don't worry about the equation. The interesting thing is this: when one plugs in the known numbers for the electron's charge, e, Planck's constant, h, and the velocity of light, c, out pops a = 1/137. There's that 137 again, a pure number.
Experimenters continued to add pieces to the Bohr model of the atom. In 1896, before the discovery of the electron, a Dutchman, Pieter Zeeman, put a Bunsen burner between the poles of a strong magnet and placed a lump of table salt in the burner. He examined the yellow light from sodium with a very precise spectrometer he had constructed. Sure enough, in the magnetic field the yellow spectral lines became broader, meaning that the magnetic field actually splits the lines. This effect was confirmed by more precise measurements up through 1925, when two Dutch physicists, Samuel Goudsmit and George Uhlenbeck, came up with a bizarre suggestion that the effect could be explained only by giving the electrons the property of "spin." In a classical object, say a top, spin is the rotation of the top around its axis of symmetry. Electron spin is the quantum analogue.
All of these new ideas, though valid by themselves, were rather ungracefully tacked on to the 1913 Bohr atomic model, like products picked up at a custom-car shop. With these accoutrements, the now greatly aggrandized Bohr theory, like an old Ford retrofitted with air conditioning, spinner hubcaps, and fake tailfins, could account for a very impressive amount of precise and brilliantly achieved experimental data.
There was only one problem with the model. It was wrong.
A PEEK UNDER THE VEIL
The crazy-quilt theory initiated by Niels Bohr in 1912 was running into increasing difficulties when a French graduate student in 1924 uncovered a crucial clue. This clue, revealed in an unlikely source, the turgid prose of a doctoral dissertation, would, in three dramatic years, yield a totally new conception of reality in the microworld. The author was a young aristocrat, Prince Louis-Victor de Broglie, sweating out his Ph.D. in Paris. De Broglie was inspired by a paper by Einstein, who in 1909 was mulling over the significance of his light quanta. How could light behave like a swarm of energy bundles—that is, like particles—and at the same time exhibit all the behaviors of waves, such as interference, diffraction, and other properties that require a wavelength?
De Broglie thought that this curious dual character of light might be a fundamental property of nature that could be applied to material objects such as electrons as well. In his photoelectric theory, following Planck, Einstein had assigned a certain energy to a quantum of light, related to its wavelength or frequency. De Brogl
ie then invoked a new symmetry: if waves can be particles, then particles (electrons) can be waves. He devised a way to assign electrons a wavelength related to their energy. His idea immediately hit pay dirt when he applied it to electrons in the hydrogen atom. An assigned wavelength gave an explanation for Bohr's mysterious ad hoc rule that only certain radii are allowed to the electron. It's totally obvious! It is? Sure. If in a Bohr orbit the electron has a wavelength of some teensy fraction of a centimeter, then only those orbits are allowed in which an integral (whole) number of wavelengths can fit around the circumference. Try this crude visualization. Go get a nickel and a handful of pennies. Place the nickel (the nucleus) on a table and arrange a number of pennies in a circle (the electron orbit) around the nickel. You'll find you need seven pennies to make the smallest orbit. This defines a radius. If you want to use eight pennies you are forced to make a bigger circle, but not any bigger circle; only one radius will do it. Larger radii will permit nine, ten, eleven, or more pennies. You can see from this dumb example that if you restrict yourself to whole pennies—or whole wavelengths—only certain radii are allowed. To get circles in between requires overlapping the pennies, and if they represent wavelengths, the waves wouldn't connect up smoothly around the orbit. De Broglie's idea was that the wavelength of the electron (the diameter of the penny) determines the allowed radii. Key to his concept was the idea of assigning a wavelength to the electron.
De Broglie, in his dissertation, speculated as to whether electrons would demonstrate other wavelike effects such as interference and diffraction. His faculty advisers at the University of Paris, though impressed by the young prince's virtuosity, were nonplused by the notion of particle waves. One of his examiners, wanting an outside opinion, sent a copy to Einstein, who wrote back this compliment about de Broglie: "He has lifted a corner of the great veil." His Ph.D. thesis was accepted in 1924 and eventually earned him a Nobel Prize, making de Broglie the only physicist up to that time to win the Prize on the basis of a dissertation. The biggest winner, though, was Erwin Schrödinger, who saw the real potential in de Broglie's work.
The God Particle: If the Universe Is the Answer, What Is the Question? Page 21
