Many towns and streets are lit at night by sodium Vapour lamps, which emit an intense, pure yellow-orange glow. Inside these lights is sodium metal. When the light is switched on, the sodium metal is slowly vaporized. As an electric current is passed through the sodium Vapour, an orange light is emitted—the same colour as the light you get when you put a small amount of a sodium compound on a spatula and place it in a Bunsen flame. Given sufficient energy (from the electric current or from a flame) sodium always emits this same wavelength of light, and it does so because of the way the electrons are arranged in a sodium atom. The energy supplied causes an electron to move from a lower energy state to a higher energy, or excited, state, and as it drops down again light is emitted. The process is a bit like a weight-lifter lifting a heavy weight—he can hold it above his head with straight arms (the excited state) but sooner or later he will drop it and the weight will fall to the ground, releasing energy with a crash, if not a broken toe. This is the origin of the lines in the atomic spectra not only for sodium but for all the elements. The flame or the electric discharge provides the energy to promote an electron to a higher energy level and, when this electron returns to its ground state, this energy is released in the form of light. If you refract the orange sodium light through a prism, you see a series of very sharp lines, with two particularly bright ones in the orange region of the spectrum at around 600 nm. Other elements produce similar spectra—even hydrogen, and since a hydrogen atom is the simplest atom of all, we shall look at the atomic spectrum of hydrogen first.

Electrons have quantized energy levels

The absorption spectrum for hydrogen was fi rst measured in 1885 by a Swiss schoolmaster, Johann Balmer, who also noticed that the wavelengths of the lines in this spectrum could be predicted using a mathematical formula. You do not need to know the details of his formula at this stage, instead let’s think about the implications of the observation that a hydrogen atom, with just one electron, has a spectrum of discrete lines at precise wavelengths. It means that the electron can only occupy energy levels with precisely determined values, in other words that the energy of an electron orbiting a proton (a hydrogen nucleus) is quantized. The electron can have only certain amounts of energy, and therefore the gaps between these energy levels (which give rise to the spectrum) likewise can only have certain well-defined values. Think of climbing a flight of stairs—you can jump up one, two, five, or even all the steps if you are energetic enough, but you cannot climb up half or two-thirds of a step. Likewise coming down, you can jump from one step to any other—lots of different combinations are possible but there is a finite number, depending on the number of steps. We mentioned an electron ‘orbiting’ a hydrogen nucleus in the last paragraph deliberately, because that is one way of thinking about an atom—as a miniature (10−23 scale!) solar system with the nucleus as the sun and the electrons as planets. This model breaks down when we look at it in detail (as we shall see shortly), but for the moment we can use it to think about why electrons must exist in quantized energy levels. To do this, we need to introduce a concept from nineteenth century physics—the experi mentally observable fact that particles such as photons and electrons can also have the char acter of a wave as well as a particle. It’s not obvious why the energy of a particle should be quantized, but it makes sense if you allow yourself to think of an electron as a wave. Imagine a taut string—a piano wire or guitar string, for example—fixed at either end. You may well know that such a string has a fundamental frequency: if you make it vibrate by hit ting or plucking it, it will vibrate in a way represented in the diagram on the right. This diagram shows a snapshot of the string; we could also represent a ‘blurred’ image of all the places you might find the string as it vibrates, such as you would get if you took a picture with a slow shutter speed. But this is not the only way the string can vibrate. An alternative possibility is shown on the right, where not only are the ends of the string stationary, but so is the point—known as a ‘node’—right in the middle. The wavelength of the vibration in this string is half that of the one above, so the frequency is double. Musically this vibration will sound an octave higher and is known as the fi rst harmonic of the fi rst vibration we showed you, the fundamental. Third and fourth possibilities for ‘allowed’ vibrations are shown below, and again these correspond musically to further harmonics of the fundamental frequency. Even if you have not met this idea in music or physics before, we hope that you can see that the vibrating string has no choice but to adopt one of these quantized frequencies—the frequency can take on only certain values because the fixed ends to the string means the wave length has to be an exact divisor of the length of the string. And as we have seen before, frequencies are directly linked to energies: the energy levels of a vibrating string are quantized. If you think of an electron as a wave, it becomes much easier to see why it can have only certain energy values. If you think of an electron orbiting a nucleus as a string looped back on itself, you can visualize from the diagrams below why only certain wavelengths are possible. These wavelengths have associated frequencies and the frequencies have associated energies: we have a plausible explanation for the quantization of the energy of an electron.