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Quantum

Neils Bohr And The "old Quantum Theory"

In the same year (1922) in which Einstein was awarded the Nobel Prize, largely for his explanation of the photoelectric effect, the Danish physicist Niels Bohr (1885–1962) received the same recognition for his quantum theory of the structure of atoms and their radiations, first put forward in 1913. Before Bohr, the quantum of action had been associated with heat radiation and light. It had also dissipated (in the hands of Einstein and others) one of Kelvin's clouds hanging over physics, that of the specific heats. Bohr seized upon the quantum to make a durable model of the atom itself.

In 1909 the New Zealander Ernest Rutherford (1871–1937), a professor in Manchester, England who was famous for his work in radioactivity, asked one of his students to study the absorption of alpha particles in a thin foil of gold. As an atom was supposed to be a sphere of diffuse positive charge with electrons stuck in it, able to vibrate in response to light or to emit light when set into vibration, Rutherford expected the alpha particle, a high speed helium nucleus, about eight thousand times heavier than the electron, to pass through with only a small deviation. Instead, many alpha particles suffered deflections by large angles, and some were even reflected in the backward direction. After verifying this amazing result and analyzing it mathematically, in 1911 Rutherford announced that the atom, about 108 cm in radius, was almost entirely empty, that the positive charge was concentrated in a region no larger than 1012 cm (the nucleus), and the electrons (79 in the case of gold) circled the nucleus as the planets circle the sun.

Niels Bohr came from a well-known academic family in Copenhagen, his father a professor of physiology and his mother belonging to a rich and cultivated Jewish family. Niels's brother Harald (1887–1951) became a famous mathematician and his son Aage (b. 1922) also won a Nobel Prize in physics. Born in 1885, Niels Bohr received his Ph.D. from the University of Copenhagen in 1911 and traveled for further study to England, first to Cambridge to work with Joseph John Thomson (1856–1940), who had discovered the electron in 1897. Thomson was responsible for the atomic model that was replaced by Rutherford's planetary picture. After a short time, Bohr moved to Manchester to work with Rutherford and began to think about atoms.

Comparing Rutherford's model of the simplest atom, hydrogen—one light electron circling a heavy positive nucleus—with a simplified solar system containing only the earth and the sun, the mechanical picture would seem to be nearly identical, with the inverse square electrostatic force in the atom replacing the inverse square gravitational force in the solar system. However, atoms that continuously interact and collide with other atoms (e.g., in a gas) remain remarkably stable, retaining their size and shape, which is not true of solar systems. Moreover, according to classical electromagnetic theory even an isolated atom would be unstable, as the electron in its orbit accelerates toward the nucleus, it should radiate energy like a small antenna. Within a small fraction of a second, the electron should spiral in toward the nucleus and be absorbed. Yet atoms appear to be almost indestructible under ordinary conditions.

In 1913 Bohr published an atomic theory that solved these difficulties, but it required the acceptance of two very new principles. The electrons revolve around the nucleus, but only in certain well-separated orbits called "stationary states." In these allowed states, the electron does not radiate, but in passing from state of energy E to another of lower energy E′ it emits a light quantum of energy hν = E − E′. Similarly it absorbs such a quantum (if present) in passing from the state of lower energy to the higher. In the case of hydrogen, the allowed energy states are circular orbits that are restricted by the condition mvr = nh/2π, where m and v are, respectively, the mass and speed of the electron and r is the radius of the circular orbit. The "quantum number" n is a positive integer. One can then show easily that the energies of the stationary states are given by En = −(1/n2) me4/2(h/2π)2 = −13.6/n2 electron volts. (The negative energy means that the electron is in a bound state. An electron with positive energy is outside of an atom.)

In 1860, Kirchhoff and Robert Bunsen (1811–1899) at the University of Heidelberg started systematic investigations of atomic spectra and showed that the frequencies of the spectral lines were characteristic of elements, even when those atoms formed part of a chemical compound. Any good theory of atomic structure would have to explain those frequencies. For the case of hydrogen, combining Bohr's two principles gave the result hv = E − E = −13.6 (1/n2 − 1/n2) electron volts. With n′ = 2, Bohr's expression gave the Balmer formula for the hydrogen lines in the visible part of the spectrum, known since 1885. Other hydrogen lines predicted to be in the ultraviolet were found, as well as those of another "one-electron" atom, namely once-ionized helium. Another success was the prediction of X-ray lines arising from heavier elements, which gave an accurate determination of the element's atomic number and hence its correct place in the periodic table of the chemical elements. This showed that there was a "missing" element, with atomic number 43, still to be discovered.

The analysis of the spectra of heavier elements was not so simple, and even the observed hydrogen spectrum showed more structure than Bohr's first model allowed. Bohr himself attacked this problem, as did others, especially Arnold Sommerfeld (1868–1951) of the University of Munich. Besides Bohr's n, the "principle quantum number," Sommerfeld introduced an "angular momentum quantum number" l, which could take on the integer values from 0 to n − 1, and a "magnetic quantum number" with integer values from − l to + l. With their help, Sommerfeld could account for many spectral lines, including the "splitting" into several components, an effect known as "fine structure." Even the hydrogen lines showed fine structure, and Sommerfeld was able to account for small relativistic effects.

Because the Bohr-Sommerfeld model had electrons following definite visualizable orbits, however restricted, including ellipses as well as circles, the model could be described as semi-classical. In the calculation of the probabilities of transitions between stationary states, which give the relative brightness of the various spectral lines, Bohr used another semiclassical idea, to which in 1923 he gave the name "correspondence principle." According to this, the classical radiation theory is valid whenever the quantum numbers have large values, so that both the frequency and the intensity of light emission is that which would arise classically from the electron's acceleration in its orbit.

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