Pass sunlight through a prism and you get a smooth rainbow. Pass the glow of hot hydrogen gas through the same prism and you get something stranger: a few sharp, bright lines of color with darkness between them. That pattern of lines is a clue to the deepest fact in this course. At the smallest scales, nature does not slide smoothly from one value to the next. It moves in steps.

Ramps and staircases

In everyday physics, quantities such as energy seem continuous, like a ramp. A swinging pendulum can have any amount of energy, and you can add or remove as little as you like. Quantum mechanics says that for a bound system, such as an electron held inside an atom, many quantities are quantized: they can take only certain separate values, like the steps of a staircase. An electron in an atom can sit on one step or the next, but never halfway between.

The first hints of this came early: Planck's packets of energy and Bohr's allowed levels. Here the aim is to see why the staircase picture matters and how it shows up everywhere once you know where to look. The story of how physicists stumbled onto it is told in QUAN 102.

Fingerprints in light

In 1814 the German optician Joseph von Fraunhofer mapped hundreds of thin dark lines crossing the spectrum of sunlight. Around 1860 Gustav Kirchhoff and Robert Bunsen showed that each chemical element gives off and absorbs light only at its own particular set of colors, so its lines act like a fingerprint. Using this method they discovered two new elements, cesium and rubidium. In 1868 astronomers studying the sun during an eclipse found a yellow line that matched no known element. The new element was named helium, after the Greek word for the sun, and it was not found on Earth until 1895.

For decades nobody knew why the lines existed. Quantum mechanics gave the answer. When an electron drops from a higher step to a lower one, the atom releases a single photon carrying exactly the energy difference between the two steps. Since the steps are fixed, only certain energy differences are possible, so only certain colors appear. A smooth ramp would produce a smooth rainbow; a staircase produces sharp lines.

Proof with electrons

Light was not the only evidence. In 1914 James Franck and Gustav Hertz fired electrons through mercury vapor and measured how much energy they lost. If atoms could absorb any amount of energy, the losses would have been spread smoothly. Instead, the electrons lost energy only in one fixed amount or not at all, and the mercury then glowed with light of the matching color. Atoms, it turned out, accept energy only in the sizes their staircase allows.

Why we never notice

If nature comes in steps, why does the world look smooth? The size of the steps is governed by a number called Planck's constant, and it is fantastically small by everyday standards. For a swinging pendulum or a rolling ball, the steps are so tiny compared with the total energy that no instrument could detect them. It is like a staircase with steps a billionth the width of a hair: from a distance it looks exactly like a ramp. Only for tiny systems, such as electrons in atoms, are the steps large compared with the energies involved.

The same tininess explains why light seems continuous. A small laser pointer releases thousands of trillions of photons every second. Our eyes and instruments average over this flood, much as a beach looks smooth from a distance even though it is made of grains.

Quantized does not mean small

A common mistake is to think that "quantum" simply means "tiny". The word comes from the Latin for "how much", and it refers to a definite amount. The key idea is discreteness, not size. Electric charge, for example, always comes in whole multiples of the electron's charge; nobody has ever found a free particle carrying half of it. Angular momentum, the measure of spinning motion, also comes in fixed units, a fact that returns in the level on spin.

Worked example: the colors of fireworks and streetlights

A fireworks designer who wants a red burst packs the shell with strontium compounds; for green, barium; for yellow, sodium; for blue, copper. When the shell explodes, the heat lifts electrons in these atoms to higher steps. As they drop back down, each element releases photons at its own fixed energies, so each burns with its own color. The designer is, in effect, choosing staircases.

The orange sodium lamps that once lit many city streets work the same way. Electricity excites sodium atoms, and nearly all their light comes out in two closely spaced yellow orange lines. That is why colors looked so strange under them: a red car looked dull brown or gray, because almost no red light was there for it to reflect. Astronomers use the identical principle in reverse. By reading the dark and bright lines in starlight, they learn which elements a star contains, how hot it is and even how fast it is moving toward or away from us, though no one will ever visit it. A pattern that puzzled scientists in 1814 now lets us take the chemical inventory of the universe from our own planet.