$$\require{cancel}$$

# 10.12 The Quantum Theory

Astronomers study large objects like planets and stars and galaxies, but their understanding depends on knowledge of the microscopic interactions of matter and radiation. For example, we have seen that temperature is a measure of the microscopic motions of atoms and molecules. The electromagnetic spectrum is our window on the universe, but to understand how this radiation is produced, we must revisit the tiny world of the atom. More than 100 years ago, scientists who studied radiation from atoms were faced with two mysteries; one was observational and the other was theoretical.

Diagram of the Electromagnetic Spectrum. Click here for original source URL

Solar spectrum showing the dark absorption lines. Click here for original source URL.

Astronomers studying the radiation from stars discovered a curious phenomenon. They noticed narrow features in visible spectra of the Sun and other stars. In addition to the smooth thermal spectrum, there were narrow, bright lines at certain wavelengths and narrow, dark lines at other wavelengths. Similar features were seen in the spectra of hot gases in the laboratory. Recall that a spectrum is a map of wavelengths, but since wavelength is inversely related to energy, a spectrum is also a map of energy. These "spectral lines" were a mystery because classical physics held that atoms could have any energy. Since energy can take any value and varies smoothly and continuously, there is no way to explain sharp features at particular energies.

Diagram of an idealizedÂ LithiumÂ atom, primarily useful to illustrate theÂ nucleusÂ of an atom. This sort of design is scientifically inaccurate in many important respects, but serves as a powerfulmandalaÂ of the nuclear age. Click here for original source URL.

Ernest Rutherford. Click here for original source URL.

The second issue was equally puzzling. Ernest Rutherford developed a model of atoms consisting of a small, dense nucleus (of protons and neutrons) surrounded by a cloud of much lighter electrons. Protons carry a positive electric charge and electrons carry an equal negative electric charge. An atom of the simplest element, hydrogen, has a single proton in the nucleus and a single orbiting electron. It seems like this orbit might continue forever. However, in the classical theory of radiation, any charged particle emits radiation if it is accelerated. (This is how we make radio waves, by sending electrons racing up and down a wire.) A simple calculation showed that the electron in a hydrogen atom should rapidly lose energy and spiral in towards the proton, pulled by the electrical attraction between the two particles. The same argument goes for any atom. Classical theory predicted that atoms should collapse in a fraction of a second, yet normal matter is obviously stable. Atoms seemed to mock physicists by their very existence!

The Bohr model for the atom, showing the nucleus and different levels that an electron can occupy. Click here for original source URL.

Neils Bohr. Click here for original source URL.

Max Planck. Click here for original source URL.

About 100 years ago, German physicist Max Planck developed a radical new theory that solved both of these mysteries. Just as matter has a fundamental unit called an atom, Planck proposed that energy has a fundamental unit called a quantum (plural: quanta). Energy is not smooth and continuous but it comes in discrete packets called photons. This revolution in physics was called the quantum theory of radiation. Neils Bohr combined Rutherford's picture of the atom — a tiny nucleus with electrons around it — with Planck's quantum theory to show how matter emits and absorbs photons. In some ways, the atom is like a tiny solar system. That is, a relatively massive central object and tiny orbiting particles. But there is an extraordinary difference. In the solar system we can imagine a planet in any orbit around the Sun; each orbit would have its own velocity and hence energy. However in an atom, electrons can occupy only specific orbits called energy levels. Physicists describe this by saying that the atom has quantized energy levels, because only certain quantities of orbital energy are possible.

Solar absorption spectrum. Click here for original source URL.

Hydrogen energy diagram. Click here for original source URL.

Iron emission spectrum. Click here for original source URL.

How does the quantum theory explain the mysteries of spectral lines and the stability of the atom? The quantum theory holds that there is a lowest energy that any electron in an atom can have. The electron can not lose energy continuously so it does not spiral in toward the nucleus and the atom does not collapse. The answer to the second mystery is particularly interesting. If an atom could gain or lose any amount of energy it could emit radiation with any wavelength. All these wavelengths would fill in to make a smooth and continuous spectrum. But electrons are held in specific energy levels. So atoms can only gain or lose the energy corresponding to an atom moving between two energy levels. This creates a set of rules for the emission and absorption of radiation in atoms.

Emission is the process where an electron loses energy in the form of a photon. Absorption is the complementary process where an electron gains energy in the form of a photon. Levels or orbits that are farthest from the nucleus have the highest energy. Each transition between levels corresponds to a photon with energy equal to the energy difference between the two levels. The classical picture corresponds to a situation where the energy levels are so finely subdivided that any energy levelis possible. In the quantum picture, the energy levels are discrete so an electron can only lose specific amounts of energy. This creates sharp lines in the spectrum since each specific energy loss corresponds to a particular wavelength of radiation. It is like the difference between sliding down a smooth slope and walking down steps. In the classical case, an electron moves to a lower energy level by losing any amount of energy. Consequently, the spectral transitions have any wavelength, and so the spectrum fills in to be smooth and featureless.

As an analogy for the quantized world of the atom, think of two different types of arithmetic. In one system you can add or subtract any fraction or decimal number. Starting with 8, say, you can add 1/2 or 1/9 or 2.37, or you can subtract 1/18 or 10.4 or 3.006. Do this many times and you will fill in the number line. This is the smooth and continuous world of classical physics. Now think of integer arithmetic. Starting with 8 again, you can add 1 or 4 or 7, or you can subtract 2 or 5 or 6. Do this many times and you will have a set of integer values. This is the quantum world. (An integer is analogous to a quantum and the number line is analogous to a spectrum of energy or wavelength.)

In the everyday world, the quantized nature of energy and matter is not apparent. Crouch down on a beach and you can see the tiny particles of rock that make up sand. From a distance, the texture appears smooth and uniform, and the structure is not visible. Let's go back to the analogy of the number line. If we step far back from the number line so that we can only see billions or trillions marked off, the individual integers are no longer visible. Seen on this large scale the integer number line appears smooth and continuous. Similarly, transactions in the everyday world involve many trillions of atoms or photons. The "graininess" of the quantum world is not visible to us.

Werner Heisenberg. Click here for original source URL.

The quantum theory of radiation may seem strange to you. It was strange and uncomfortable for many physicists too! The theory became accepted because it describes how nature works very well. In the 1930s, physicists were still reeling from the idea of the quantum when German physicist Werner Heisenberg came up with an equally bizarre idea. He showed that quantities like energy, momentum, and position can not be defined with absolute precision. This idea is enshrined in physics as the Heisenberg uncertainty principle. Uncertainty is not the best word to describe this principle; its better thought of as imprecision. Nature puts a floor on the precision with which we can make measurements. In particular, the product of the uncertainty in jointly measuring position and momentum is limited to Planck's constant: Δx \$times; Δp > h/4π. Similarly, the product of the uncertainty in measuring energy and time is limited to Planck's constant: ΔE × Δt > h/4π. Since h is a tiny number this imprecision is not noticeable in the everyday world.

Scientists believe that the solar system analogy of the atom, with particles like hard balls in specific orbits, is a weak one. In the quantum model of the atom, the electron is not a discrete object like a billiard ball. It has a certain probability of being at various positions, so the electron is not just at one position in its orbit like a planet would be. The imprecision described by Heisenberg is only substantial on the scale of atoms or subatomic particles. It does not limit our knowledge of the everyday world. The "fuzziness" of the subatomic world is not visible to us.

Light can be thought of as either a wave or a particle. In the quantum view of the subatomic world, a particle can also be thought of as a wave. Let's look at the difference between the classical and quantum ways of looking at particles and waves. The classical view of a particle is something hard-edged like a marble or a ball bearing. The quantum view of the particle is somewhat fuzzy, because it has a probability of being found over a range of space. We can make the same comparison for a wave. A classical wave extends off forever into space in either direction. The quantum view considers the energy of the wave to be more concentrated in space. In the classical view, particles and waves appear to be totally distinct entities. In the quantum view, they are quite similar!

There are two alternative views of electrons in an atom. The classical model of a miniature solar system does not explain why the electrons are confined to certain orbits. But the quantum model is based on the wave properties of particles. A useful analogy for imagining the quantized energy levels of an atom is to think of the set of vibration modes of a plucked string. The vibrations of a guitar string are made of a whole number of waves. In the same way, each energy level corresponds to a whole number (as opposed to a fraction) of waves and the orbits therefore have quantized wavelengths (or energies). The guitar string analogy is also appropriate because the wave's energy is distributed in space, not concentrated like a hard-edged particle. The quantum idea of particles as waves may seem counter-intuitive, but it helps explain why the atom should have quantized energy levels.