Bunch-Hellemans, The quantum

One of the most important tools of science had its start in 1665 or 1666 when young Isaac Newton observed that a prism demonstrates that white light is a mixture of the colors of the rainbow. Although scientists realized earlier that the rainbow is formed when light is broken into different colors (the spectrum), Newton was the first to make a thorough study of the subject. Later, other scientists investigated this effect for light produced by heating different elements; from this investigation we learned of the composition of the stars. It is well known that study of the spectrum reveals the composition of a heated body or gas. Much less well known is that this study also reveals much about the atom.

The spectrum not only appears in the rainbow, it is hidden in another effect. An observation that must have been made in antiquity is that as iron is heated to forge it, it first becomes dull red, then brighter red, and gradually white. Other solid materials that do not burn behave much the same way. After formulation of the electromagnetic theory of light, theorists tried to explain this phenomenon from first principles. It was apparent that longer wavelengths appear at moderate temperatures. As temperatures rise, shorter and shorter wavelengths are emitted. When the material becomes white hot, all the wavelengths are represented. Studies of even hotter bodies — stars — showed that in the next stage the longer wavelengths drop out, so that the color gradually moves toward the blue part of the spectrum.

Efforts to make theoretical sense of the way the spectrum gradually appears and disappears were at first unsuccessful. For one thing, theory suggested that a perfectly black body — one that absorbs every wavelength of electromagnetic radiation equally well — would, upon heating, radiate every wavelength equally well. Experiments with simulated black bodies, however, showed that they behave in the same way that iron does when it is heated. In the 1890s Wilhelm Wien and Lord Rayleigh each tried to find a formula to explain these phenomena, but each failed in a different way. Wien’s formula worked near the blue end of the spectrum and above, but failed for long wavelengths. Rayleigh’s formula was just the opposite, good for long wavelengths and not for short.

In 1900 Max Planck found an explanation that worked for all wavelengths, but little attention was paid to it. Planck made the assumption — which seemed quite odd at the time, even to Planck — that electromagnetic radiation could only be emitted in packets of a definite size, which he called quanta. People took notice of Planck’s quantum only when Einstein, in 1905, used the idea to explain the photoelectric effect, to reconcile theory and experiment for heat, and to account for the propagation of light without relying on an “ether.” It appeared that Planck’s quanta went beyond theory and had a physical reality.

By 1911 Ernest Rutherford had established that the atom has a positive nucleus surrounded by orbiting electrons. Like the blackbody problem, however, the theory of the atom did not match experiment. Electrons orbiting a nucleus should give off radiation constantly, resulting in the electron falling into the nucleus. But atoms do not give off that kind of energy and they are usually quite stable. Niels Bohr turned to Planck’s quantum to salvage the theory.

The size of the quantum, based on a pure number called Planck’s constant, could be calculated. Starting in 1913, Bohr calculated the quantum of the simplest case, hydrogen, in which a single electron orbits a proton. He showed that experiment and theory could be reconciled by saying that the quantum restricts the electron to particular orbits. For each counting number (1, 2, 3, . . .) there is one permissible orbit. For a given electron, the orbit it was in could be assigned that number, called its quantum number. Bohr based his calculations on the lines that form the spectrum of hydrogen gas (when a pure gas is heated, the spectrum consists of discontinuous lines, not a full rainbow). Bohr explained the lines by saying the light is emitted when the electron changes from a higher quantum number to a lower one.

Hydrogen does not display a continuous spectrum because the electron moves from orbit to orbit in “quantum jumps.” More complex atoms were beyond direct calculations, but approximations indicated that the same approach was correct.

But there were minor complications. Bohr’s model explained the large lines in the spectrum, but these lines are broken into smaller lines, called the fine structure of the spectrum. In 1915 Arnold Sommerfeld introduced a second quantum number to explain the fine structure. This was based on the idea that orbits allowed to electrons are ellipses, not circles. Next, it was observed that since the spectrum is affected by a magnet — the Zeeman effect — there needs to be a third quantum number to account for the magnetic state of the electron. Finally, in 1925 George Uhlenbeck and Samuel Goudsmit found that electrons spin, necessitating yet another quantum number. Each number is an integer that describes the specific state of the electron. If you know that the numbers are, say, 3, 1, 1, 2, then you have a precise description of the electron in its orbit.

The discovery of spin was a major breakthrough, resulting in the rapid development of what is now known as the quantum theory. In 1925 Wolfgang Pauli determined that four quantum numbers is just right. Everything known about an electron in an atom can be reduced to the four numbers. Furthermore, no two electrons in an atom can have the same numbers. This Pauli exclusion principle, as it became known, accounts for how electrons are arranged in all atoms and tells why sulfur has different properties than tin.

About the same time, Werner Heisenberg found that arrays of quantum numbers could be used to calculate lines in the spectrum. This method of calculation is called the Heisenberg matrix mechanics. Earlier, Louis de Broglie had proposed that every particle has a wave associated with it. Erwin Schrödinger used de Broglie’s idea to calculate the spectral lines. Later it was shown that the Heisenberg matrix mechanics and the Schrödinger wave equation are equivalent.

In 1927 Heisenberg put forward the idea that it is theoretically impossible to determine the position and the momentum of an electron at the same time. The greater the degree of accuracy about one quantity the less the accuracy of the other. This uncertainty principle, as it is known, was later extended to other particles and other quantities. Max Born suggested that the Schrödinger equation could be interpreted as giving the probability that an electron is located in a particular orbit. This interpretation is still the most common in quantum theory.

Although Schrödinger’s wave equation gives good results, they are not perfect. The wave equation does not take spin or the theory of relativity into effect. In 1928 Paul Dirac revised the equation to include spin and relativity. Dirac’s theory was important because it revealed for the first time the existence of antimatter, but the mathematics is so complicated that physicists still use the Schrödinger wave equation. Dirac’s theory essentially completed classical quantum theory. After World War II, physicists developed quantum electrodynamics, a method of calculating the behavior of electrons and other particles that is even more precise than classical quantum theory.