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Section 4: Mysteries of Matter

Early in the 20th century, it was known that everyday matter consists of atoms and that atoms contain positive and negative charges. Furthermore, each type of atom, that is, each element, has a unique spectrum—a pattern of wavelengths the atom radiates or absorbs if sufficiently heated. A particularly important spectrum, the spectrum of atomic hydrogen, is shown in Figure 12. The art of measuring the wavelengths, spectroscopy, had been highly developed, and scientists had generated enormous quantities of precise data on the wavelengths of light emitted or absorbed by atoms and molecules.

The spectrum of atomic hydrogen.

Figure 12: The spectrum of atomic hydrogen.

Source: © T.W. Hänsch. More info

In spite of the elegance of spectroscopic measurement, it must have been uncomfortable for scientists to realize that they knew essentially nothing about the structure of atoms, much less why they radiate and absorb certain colors of light. Solving this puzzle ultimately led to the creation of quantum mechanics, but the task took about 20 years.

The nuclear atom

In 1910, there was a major step in unraveling the mystery of matter: Ernest Rutherford realized that most of the mass of an atom is located in a tiny volume—a nucleus—at the center of the atom. The positively charged nucleus is surrounded by the negatively charged electrons. Rutherford was forced reluctantly to accept a planetary model of the atom in which electrons, electrically attracted to the nucleus, fly around the nucleus like planets gravitationally attracted to a star. However, the planetary model gave rise to a dilemma. According to Maxwell's theory of light, circling electrons radiate energy. The electrons would generate light at ever-higher frequencies as they spiraled inward to the nucleus. The spectrum would be broad, not sharp. More importantly, the atom would collapse as the electrons crashed into the nucleus. Rutherford's discovery threatened to become a crisis for physics.

The Bohr model of hydrogen

Niels Bohr, a young scientist from Denmark, happened to be visiting Rutherford's laboratory and became intrigued by the planetary atom dilemma. Shortly after returning home Bohr proposed a solution so radical that even he could barely believe it. However, the model gave such astonishingly accurate results that it could not be ignored. His 1913 paper on what became known as the "Bohr model" of the hydrogen atom opened the path to the creation of quantum mechanics.

Bohr's model of an atom.

Figure 13: Bohr's model of an atom.

Source: © Daniel Kleppner. More info

Bohr proposed that—contrary to all the rules of classical physics—hydrogen atoms exist only in certain fixed energy states, called stationary states. Occasionally, an atom somehow jumps from one state to another by radiating the energy difference. If an atom jumps from state b with energy Eb to state a with lower energy, Ea, it radiates light with frequency given by . Today, we would say that the atom emits a photon when it makes a quantum jump. The reverse is possible: An atom in a lower energy state can absorb a photon with the correct energy and make a transition to the higher state. Each energy state would be characterized by an integer, now called a quantum number, with the lowest energy state described by n = 1.

Bohr's ideas were so revolutionary that they threatened to upset all of physics. However, the theories of physics, which we now call "classical physics," were well tested and could not simply be dismissed. So, to connect his wild proposition with reality, Bohr introduced an idea that he later named the Correspondence Principle. This principle holds that there should be a smooth transition between the quantum and classical worlds. More precisely, in the limit of large energy state quantum numbers, atomic systems should display classical-like behavior. For example, the jump from a state with quantum number n = 100 to the state n = 99 should give rise to radiation at the frequency of an electron circling a proton with approximately the energy of those states. With these ideas, and using only the measured values of a few fundamental constants, Bohr calculated the spectrum of hydrogen and obtained astonishing agreement with observations.

Bohr understood very well that his theory contained too many radical assumptions to be intellectually satisfying. Furthermore, it left numerous questions unanswered, such as why atoms make quantum jumps. The fundamental success of Bohr's model of hydrogen was to signal the need to replace classical physics with a totally new theory. The theory should be able to describe behavior at the microscopic scale—atomic behavior—but it should also be in harmony with classical physics, which works well in the world around us.

Matter waves

By the end of the 1920s, Bohr's vision of a new theory was fulfilled by the creation of quantum mechanics, which turned out to be strange and even disturbing.


Figure 14: This diffraction pattern appeared when a beam of sodium molecules encountered a series of small slits, showing their wave-like nature.

Source: © D.E. Pritchard. More info

A key idea in the development of quantum mechanics came from the French physicist Louis de Broglie. In his doctoral thesis in 1924, de Broglie suggested that if waves can behave like particles, as Einstein had shown, then one might expect that particles can behave like waves. He proposed that a particle with momentum p should be associated with a wave of wavelength = h/p, where, as usual, h stands for Planck's constant. The question "Waves of what?" was left unanswered.

de Broglie's hypothesis was not limited to simple particles such as electrons. Any system with momentum p, for instance an atom, should behave like a wave with its particular de Broglie wavelength. The proposal must have seemed absurd because in the entire history of science, nobody had ever seen anything like a de Broglie wave. The reason that nobody had ever seen a de Broglie wave, however, is simple: Planck's constant is so small that the de Broglie wavelength for observable everyday objects is much too small to be noticeable. But for an electron in hydrogen, for instance, the deBroglie wavelength is about the size of the atom.


Figure 15: An interference pattern builds up as individual electrons pass through two slits.

Source: © Reprinted courtesy of Dr. Akira Tonomura, Hitachi, Ltd., Japan. More info

Today, de Broglie waves are familiar in physics. For example, the diffraction of particles through a series of slits (see Figure 14) looks exactly like the interference pattern expected for a light wave through a series of slits. The signal, however, is that of a matter wave—the wave of a stream of sodium molecules. The calculated curve (solid line) is the interference pattern for a wave with the de Broglie wavelength of sodium molecules, which are diffracted by slits with the measured dimensions. The experimental points are the counts from an atom (or molecule) detector. The stream of particles behaves exactly like a wave.

The concept of a de Broglie wave raises troubling issues. For instance, for de Broglie waves one must ask: Waves of what? Part of the answer is provided in the two-slit interference data in Figure 15. The particles in this experiment are electrons. Because the detector is so sensitive, the position of every single electron can be recorded with high efficiency. Panel (a) displays only eight electrons, and they appear to be randomly scattered. The points in panels (b) and (c) also appear to be randomly scattered. Panel (d) displays 60,000 points, and these are far from randomly distributed. In fact, the image is a traditional two-slit interference pattern. This suggests that the probability that an electron arrives at a given position is proportional to the intensity of the interference pattern there. It turns out that this suggestion provides a useful interpretation of a quantum wavefunction: The probability of finding a particle at a given position is proportional to the intensity of its wavefunction there, that is to the square of the wavefunction.