Topic 3.12: Properties of Photons

Properties of photons: the energy in one packet

Now back to the mystery on the mountain, where a green laser, with more energy in every photon, did nothing at all. To solve it, we need to know exactly how much energy one photon carries. The wavelength, lambda, is the distance from one crest to the next, and the frequency, nu, is how many crests go past one spot every second.

Make the wave shorter and more crests go by each second, stretch it out and fewer go by, because every color travels at the same speed. That speed is c, about three hundred million meters a second, so wavelength times frequency always equals c. A photon’s energy depends on just one thing, its frequency, and the rule is E equals h nu, where h is Planck’s constant, and both of these equations are on the AP equation sheet.

So let’s find the energy of one photon from the laser. Five hundred eighty-nine nanometers, times ten to the minus nine meters per nanometer, is five point eight nine times ten to the minus seven meters.

The frequency is c over lambda, the meters cancel, and that’s five point oh nine times ten to the fourteenth per second, about five hundred trillion crests every second. Multiply by Planck’s constant and the seconds cancel, so one photon carries three point three seven times ten to the minus nineteen joules, with three significant figures, like the wavelength. That’s for one photon, which is why it’s so tiny, and a whole mole of them carries about two hundred kilojoules.

Put the two equations together, swapping nu for c over lambda, and E equals h c over lambda, so a shorter wave means a bigger photon, from red, the smallest here, up to blue, the biggest. Now you fill in the blank in that strip. The green laser has a wavelength of five hundred thirty-two nanometers, so what’s the energy of one of its photons, in joules?

Exactly the right amount

So why won’t sodium take the green photon? Here are the lowest energies a sodium atom is allowed to have, with nothing in between, and right now it’s on the lowest step. A green photon’s energy overshoots the first step, so the atom lets it fly straight past, and a red photon falls short and passes through too. It’s like a vending machine that only takes exact change, so too little won’t work, and neither will too much.

Tune the photon until this sodium atom takes it. When the energy matches, the photon disappears completely, and the atom’s energy goes up by exactly three point three seven times ten to the minus nineteen joules, the energy of that one photon.

About sixteen billionths of a second later it drops back down and gives off a brand-new photon with that same energy, in a random direction, and the few that happen to head down toward the telescope are the artificial star. Only a photon big enough to rip the electron out completely plays by different rules, so for a jump between two levels, the match has to be exact.

Back on the mountain, each beam is a stream of these photons, and each laser puts out twenty-two watts, which is twenty-two joules every second, so dividing by the energy of one photon gives about six and a half times ten to the nineteenth photons every second. Here are two dials, the laser’s color and its power. Make the star brighter without changing the energy of a single photon.

So when the atom absorbs a photon, its energy changes by plus three point three seven times ten to the minus nineteen joules, and when it emits one, by minus that same amount. The photon’s energy is always the positive size of the gap, and the plus or minus sign belongs to the atom, telling you whether it gained energy or lost it.

From an energy gap to a color

Molecules follow exactly the same rule. Remember the blue sports-drink dye we held on to? It absorbs best at six hundred thirty nanometers, which is orange-red. Try the energy of one of those photons with me, using h c over lambda in one step, so that’s six point six two six times ten to the minus thirty-fourth, times two point nine nine eight times ten to the eighth, over six point three oh times ten to the minus seven meters.

It’s three point one five times ten to the minus nineteen joules, which matches one of the dye’s electronic steps. If someone hands you the step instead and asks for the color, flip it around, so lambda equals h c over E. The units check out, because joule-seconds times meters per second, divided by joules, leaves just meters.

Now let’s keep the promise from the start, with the kind of question the AP exam likes. These are sodium’s real energy levels, negative because they’re measured from zero, where the electron would be free. What wavelength, in nanometers, does the laser need to lift the atom from the lower level to the upper one?

So let’s go back to the mountain and dial in the wavelength you just worked out, five hundred eighty-nine nanometers, and there they are, four brand-new stars, switched on by your number. The telescope locks onto your stars, the mirror starts reshaping itself, and the blur just melts away.