By Joshua N. Winn, Princeton University
While observing the solar spectrum, along with the continuous ribbon of color, one can also see dark lines. Interestingly, sunlight is missing certain colors. For example, there’s a conspicuous line in the red part of the spectrum, and another one in the yellow. Why is that?
The Spectrum of Sunlight
The pattern of lines, in the spectrum of sunlight, might look sort of random, but it is a code. It contains a message—from the atoms and ions in the Sun’s outer layers. They broadcast information about their temperature, abundance, and much more. The dark lines were first observed in 1802. The message, however, wasn’t decoded until the 1920s, as it was written in the language of quantum theory, which wasn’t developed until the 1920s.
Setting aside quantum physics, we can imagine an atom as a tiny solar system in which the attraction is provided by the electrical force, instead of gravity. Electrons orbit the nucleus, just as planets orbit the Sun. The classical theory of electromagnetism says any accelerating charge will radiate. If a charge suddenly starts moving, its electric field develops a pattern of kinks, abrupt changes in direction, that propagate outward at the speed of light. Well, an electron orbiting a nucleus is also accelerating: a centripetal acceleration. It feels an inward force. That’s what keeps it bound to the nucleus. So, an orbiting electron should emit electromagnetic waves, which carry away energy.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Gravitational Waves Irrelevant to Planetary Motion
In the case of planets, the total orbital energy—kinetic plus gravitational potential—is −GMm/2a, where a is the orbital distance. A similar formula applies to the case of an electron orbiting a proton under the influence of the electrical attraction: energy is −eta(e2/2a), where eta is the Coulomb constant, and the little e signifies the magnitude of the charge on either the proton and the electron. If the system is gradually losing energy, by radiating electromagnetic waves, E is getting lower, it’s becoming more negative. That means a must be shrinking. The electron should spiral inward toward the nucleus.
That should happen to planets, too. An accelerating planet produces gravitational waves that fly away at the speed of light. In principle, this should cause planets to spiral inward and crash into the Sun. But the timescale over which the orbit shrinks is orders of magnitude longer than the age of the universe, which makes gravitational waves irrelevant to planetary motion.
But electromagnetism is much stronger than gravity, and atoms are much smaller than planetary systems. And when one calculates how long it should take for an electron to spiral inward and crash into the nucleus, it comes out to be on the order of 10 nanoseconds. All the atoms in the universe should collapse within 10 nanoseconds.
Electron, a Trapped Wave
So then, there must be something seriously wrong with the classical model of the atom. What’s wrong, of course, is that it ignores quantum theory. Electrons are not just particles. They have wavelike properties, too. An electron in an atom is like a wave that’s trapped near the nucleus.
Now, think about what happens to sound waves when they’re trapped in place, a so-called standing wave. Like a vibrating piano string. Or the column of air in an organ pipe. The head of a kettledrum. In all those cases, strong vibrations only happen at certain frequencies. The only wave patterns on a piano string that lead to sustained vibrations are the ones where the nodes—the points of zero motion—coincide with the ends, where the string is held fixed. There’s a discrete spectrum of wavelengths: not continuous. That’s what happens when we have a standing wave.
The Schrödinger Equation
Coming back to the atom, because the electron’s wave nature, its wave function, is confined by the electrical attraction to the nucleus, it, too, is a standing wave, and the possibilities for its energy are restricted to a discrete set. Those energies are found by solving an equation—the time-independent Schrödinger equation. One gets a discrete set of energies, instead of a continuum, which means there’s some lowest-energy state—a minimum orbital distance—which we call the ground state. In the context of gravity, that would be like a certain distance from the Sun, inside of which, planets can’t exist.
For hydrogen, the energy levels of the electron obey a simple equation: −13.6 electron volts over n2, where n is a whole number. The ground state is −13.6 eV. This agrees with the order-of-magnitude calculation. One can represent these levels on an energy diagram, in which height is proportional to energy. Because the energies vary as 1 over n2, they bunch up near zero as n increases.
For bigger atoms with more electrons, and for molecules, with more than one nucleus, the energy diagrams are more complicated, but the point remains that the energy levels are discrete, and there’s a ground state.
This explains why atoms and molecules are stable: the electrons in the ground state can’t radiate. To do so they would need to lose energy, and they’re already in the minimum energy state. And it also explains the dark lines in the Sun’s spectrum. The thing is, an electron can jump from one level to a higher one, but only if it absorbs just the right amount of energy. This is similar to the energy from a passing photon whose energy is equal to the difference between the two energy levels.
Common Questions about the Solar Spectrum
The pattern of lines might look sort of random, but it is a code. It contains a message from the atoms and ions in the Sun’s outer layers. They broadcast information about their temperature, abundance, and much more.
An accelerating planet produces gravitational waves that fly away at the speed of light. In principle, this should cause planets to spiral inward and crash into the Sun.
Electrons are not just particles. They have wavelike properties, too. An electron in an atom is like a wave that’s trapped near the nucleus.
