Light spreads like a wave but delivers its energy in lumps, one photon at a time, each carrying an amount set only by its frequency. This chapter shows the experiments that forced physics to accept that, and why it explains everything from sunburn to lasers to why your Wi-Fi is harmless.
Common mix-up: brighter does not mean more energetic photons. Brightness is how many photons arrive each second; color, meaning frequency, is how much energy each one carries. A floodlight of red light can’t do what a single ultraviolet photon does, which is why a glowing stove won’t give you a tan and a few hours of summer sun will.
By 1900 physicists could measure the glow of a hot oven very precisely, and they could not explain it. The classical theory treated the light inside the oven as waves that each share in the heat equally. There are more possible short waves than long ones, so the theory predicted the glow should climb without limit toward the ultraviolet, a total of infinite energy. Real ovens obviously don’t do that. The failure was later nicknamed the ultraviolet catastrophe.
In December 1900 Max Planck found a formula that matched the measurements perfectly. To derive it he had to assume something he didn’t believe: that the oven walls could only trade energy with light of frequency f in whole lumps of size hf. He later called it “an act of desperation.”
The lumps tame the curve. At low frequency each lump is tiny compared with the thermal energy available, so light behaves classically. At high frequency each lump costs more than the walls can easily supply, so short-wavelength light is rarely made at all, and the glow falls away instead of exploding. That is the curve you can play with in Chapter 05’s blackbody instrument.
h is minuscule, 6.626 × 10⁻³⁴ joule-seconds, which is why nobody had noticed. A green photon carries 3.7 × 10⁻¹⁹ J. Lifting a 1 g paper clip by 1 mm takes about 10⁻⁵ J, the energy of some 26 trillion green photons.
Classically, every possible wave pattern (mode) in the oven holds an average energy kBT, with kB = 1.381 × 10⁻²³ J/K. The number of modes per unit frequency grows as f², so the energy per unit frequency grows as f² without bound: the Rayleigh–Jeans law.
If a mode can hold only 0, hf, 2hf, 3hf… then Boltzmann statistics give an average of hf / (ehf/kBT − 1) instead. When hf ≪ kBT this is ≈ kBT, the classical answer. When hf ≫ kBT it collapses exponentially, because even one lump is unaffordable. Multiply by the f² mode count and you get Planck’s curve.
The crossover, hf = kBT, sits at 6.25 THz for room temperature (300 K), in the far infrared. Below it, lumpiness hardly matters; above it, it dominates.
In 1887 Heinrich Hertz, the man who first made radio waves on purpose, noticed that ultraviolet light helped sparks jump a gap. Light was knocking electrons out of the metal. In 1902 Philipp Lenard measured how, and the results made no sense for a wave.
A wave delivers energy smoothly, in proportion to its intensity. So brighter light should shake electrons harder and eject them faster, and even dim light of any color should eventually pile up enough energy to free one. Lenard found three things instead:
In 1905 Albert Einstein proposed the simple answer: light itself comes in lumps. Each photon gives all its energy to a single electron, or none. To escape, the electron has to pay an exit fee, the work function φ, which is about 2.1 eV for cesium and 4.3 eV for zinc. Whatever is left becomes kinetic energy. A photon below the fee can’t free anyone, and a million such photons can’t either, because they don’t pool their energy.
Robert Millikan spent a decade trying to disprove this. His 1916 measurements confirmed it instead: kinetic energy plotted against frequency gave straight lines with the same slope for every metal, and that slope matched Planck’s h to within about 1%. Einstein’s 1921 Nobel Prize was awarded specifically for the photoelectric law, not relativity.
Treat light as a smooth wave and ask how long one atom takes to soak up enough energy. Take a dim beam of 0.01 W/m², about a hundred-thousandth of sunlight, and let an atom collect from an area of roughly (0.1 nm)² = 10⁻²⁰ m². It gathers 10⁻²² W. Freeing an electron with 2 eV (3.2 × 10⁻¹⁹ J) would take 3,200 seconds, nearly an hour.
Experiments see electrons within a few billionths of a second, a limit already measured in 1928. The energy is not trickling in. It arrives all at once, in one photon, at one place.
The cleanest measurement is the stopping voltage: put a reverse voltage on a collector and raise it until even the fastest electrons can’t reach it. Then eVstop = KEmax, so a voltmeter reads the electron energy directly in volts. Plot Vstop against f and the slope is h/e.
Pick a metal, then slide the wavelength and the brightness. Watch whether electrons come out, and how fast. Notice that brightness never rescues a color below threshold.
Because one photon interacts with one molecule at a time, what light can do depends on the energy per photon, not on the total power. Physicists measure it in electron volts: 1 eV is the energy an electron gains crossing 1 volt, 1.602 × 10⁻¹⁹ J. Lay the spectrum out in those units and it falls into clear zones.
Radio and microwaves carry almost nothing per photon. A 2.4 GHz Wi-Fi photon has about 10⁻⁵ eV, a hundred-thousandth of an electron volt. Even the random thermal jiggling of molecules at room temperature, about 0.026 eV, is 2,600 times bigger. These photons can push electrons around in an antenna and warm water, but they can’t change a single chemical bond.
Infrared photons, 0.001 to 1.7 eV, match molecular vibrations. They warm things by shaking bonds, which is why you feel a fire’s infrared as heat.
Visible light, about 1.7 to 3.3 eV, is the first band that matches the steps of outer electrons in molecules. Those steps are what chemical bonds are made of, a few eV each: a carbon–carbon bond holds about 3.6 eV, an oxygen–hydrogen bond about 4.8 eV. So a single visible photon can flip the retinal molecule in your eye or drive a step of photosynthesis. Your rod cells can respond to a single photon.
Ultraviolet photons, from 3.3 eV up, start breaking bonds outright. UV-B and UV-C photons are absorbed by DNA and can fuse neighboring bases together. That damage, not heat, is sunburn. Past about 10 eV (wavelengths below 124 nm), one photon can strip an electron from most atoms: hydrogen needs 13.6 eV, water about 12.6 eV. That is ionizing radiation, and X-rays (thousands of eV) and gamma rays (millions) are deep inside it.
“Photons don’t pool their energy” is true at ordinary brightness. It fails at extreme intensity. If photons arrive densely enough, a molecule can absorb two or more within the fleeting lifetime of an intermediate state, about 10⁻¹⁵ s, and add their energies.
Two-photon microscopes use this to excite dyes with infrared light only at the focus, where the intensity is enormous, giving sharp 3-D images deep in tissue. Ultrafast lasers can ionize air outright, with several infrared photons per electron.
The required intensities are around 10¹⁰ to 10¹⁴ W/cm², squeezed into femtosecond pulses. No radio transmitter, router or lamp comes within many orders of magnitude, which is why the single-photon rule is the right rule for everyday radiation safety.
Divide a beam’s power by the energy of one photon and you get the number of photons per second. The numbers explain why the lumps went unnoticed for so long.
A 1 mW red laser pointer at 650 nm emits photons of 3.06 × 10⁻¹⁹ J each, so 10⁻³ ÷ 3.06 × 10⁻¹⁹ ≈ 3.3 × 10¹⁵ photons per second. A Wi-Fi router transmitting 100 mW at 2.4 GHz uses photons 192,000 times weaker, about 1.6 × 10⁻²⁴ J, so it sends roughly 6.3 × 10²² per second. A 50 kW FM station at 100 MHz: about 7.5 × 10²⁹.
With numbers like that, the arrival of individual lumps averages into a perfectly smooth wave. That is why radio engineering never needs photons: radio photons are so cheap and so numerous that the classical wave description is exact for every practical purpose. Graininess only shows when light is very dim or photons are very energetic, as in a Geiger counter’s clicks or a camera sensor’s noise in the dark.
Photons also carry momentum, even though they have no mass: p = h/λ. A single green photon at 532 nm carries 1.25 × 10⁻²⁷ kg·m/s. Absorb a beam of power P and it pushes with force P/c; reflect it and the push doubles, because the photons bounce back.
For sunlight at Earth, 1,361 W/m², that is 4.5 µN/m² on a black surface and 9.1 µN/m² on a perfect mirror. A mirror sail the size of a tennis court, about 261 m², feels 2.4 mN, the weight of a quarter of a gram. Tiny, but it never stops pushing and needs no fuel. Japan’s IKAROS (2010) and The Planetary Society’s LightSail 2 (2019) both steered on it, and the same push sweeps comet dust into tails that point away from the Sun.
In 1923 Arthur Compton scattered X-rays off electrons in graphite and found the scattered X-rays had longer wavelengths, by an amount that depended only on the scattering angle θ: Δλ = (h / mec)(1 − cos θ).
That is exactly what you get by treating the X-ray as a particle with energy hf and momentum h/λ colliding elastically with an electron, conserving both. The constant h/mec, the Compton wavelength of the electron, is 2.426 pm. For visible light (around 500,000 pm) a shift of a few pm is undetectable; for a 70 pm X-ray it is several percent, plain to see.
Compton scattering is now how gamma rays mostly lose energy in the body, and the reason radiotherapy dose builds up below the skin surface rather than at it.
Set any power and any frequency, or pick a real source. The bar shows where the photon count lands, from countable clicks to an unbroken classical wave.
Send light through two narrow slits close together and onto a screen, and you get stripes, bright and dark: interference fringes. Where the waves from the two slits arrive in step they add; where one arrives a half-wave behind the other they cancel. The stripes are spaced by λL/d. For green laser light (532 nm), slits 0.25 mm apart and a screen 1 m away, that’s 2.1 mm. This is Thomas Young’s experiment from 1801, and for a century it was the proof that light is a wave.
Now turn the light down until only one photon is in the apparatus at a time. G. I. Taylor did this in 1909 with light so feeble that his longest exposure took about three months. Each photon arrives at the screen as a single dot, in one spot, like a particle. Where any one dot lands is unpredictable. But after thousands of dots, the fringes appear, exactly as before.
So each photon must somehow pass through both slits as a wave, interfere with itself, and then deliver its energy at one point as a particle. The wave tells you the probability of finding the photon at each spot: bright fringes are likely landing places, dark fringes are places it never lands. If you add a detector that reveals which slit each photon went through, the fringes vanish and you get two plain blobs. You can learn the path or see the interference, never both.
It isn’t only light. Electrons, neutrons, atoms and even big molecules make the same fringes, with a wavelength set by their momentum, λ = h/p (Louis de Broglie, 1924). An electron accelerated through 100 V has a wavelength of 0.12 nm, about an atom’s width, which is why electron microscopes can see atoms. Akira Tonomura’s team filmed electrons building up fringes one by one in 1989.
Quantum mechanics assigns each route a complex number, an amplitude ψ. For two open slits the amplitude at the screen is ψ₁ + ψ₂, and the probability of a photon landing there is |ψ₁ + ψ₂|² = |ψ₁|² + |ψ₂|² + 2|ψ₁||ψ₂| cos Δφ. That last cross term is the interference; Δφ is the phase difference between the routes, 2π d y / (λ L) for small angles.
If anything in the universe records which slit was taken, the two routes no longer lead to the same final state, the cross term averages to zero, and you get |ψ₁|² + |ψ₂|²: two blobs, no fringes. Nothing has to “disturb” the photon in a crude mechanical way; distinguishability alone is enough.
Each slit also has a width a, which spreads its light into a broad single-slit envelope, sinc²(π a y / λL). The fringes ride inside that envelope, as you’ll see in the instrument when you close one slit.
Each dot is one photon detected on the screen, 1 m behind two slits each 40 µm wide. Change the wavelength or slit spacing (which clears the screen), speed up the arrivals, or close one slit and watch the fringes disappear.
Atoms can only hold energy in fixed amounts, because their electrons can only occupy certain standing-wave patterns. In hydrogen the allowed energies are −13.6 eV ÷ n², for n = 1, 2, 3… (Niels Bohr, 1913). An electron dropping from one level to a lower one emits a photon carrying exactly the difference, so its color is fixed.
Drops that end on level 2 give hydrogen’s visible Balmer series: 3→2 is 1.89 eV, red light at 656 nm; 4→2 gives 486 nm (blue-green), 5→2 gives 434 nm (blue), 6→2 gives 410 nm (violet). Johann Balmer found the pattern in these four numbers in 1885 without knowing why it worked. Shine white light through cool hydrogen and the reverse happens: the same four colors are absorbed, leaving dark lines.
Every element has its own set of lines, a barcode. Joseph von Fraunhofer catalogued hundreds of dark lines in sunlight in 1814. In 1868 astronomers found a yellow line at 587.6 nm in the Sun matching no known element and named the element helium after the Greek for Sun; it wasn’t found on Earth until 1895. Line spectra are how we know what stars are made of, how fast they move (the lines shift by the Doppler effect), and how hot they are.
A laser runs this in a special way. Einstein showed in 1917 that a passing photon of exactly the right energy can stimulate an excited atom to emit a twin photon: same frequency, same direction, same phase. Pump more atoms into the upper level than the lower (a population inversion), put mirrors at both ends so photons pass back and forth, and each pass multiplies them. One partly transparent mirror lets out a beam of light that is all one color and marching in step. Charles Townes built the microwave version, the maser, in 1954; Theodore Maiman made the first laser, a ruby rod glowing at 694.3 nm, on 16 May 1960.
With only two levels, pumping can at best make the populations equal. Then every photon is as likely to be absorbed as to stimulate an emission, and there is no gain. That’s why the very first proposals looked hopeless.
A three-level laser such as ruby pumps atoms (with a flash lamp) up to a short-lived level that quickly drains into a long-lived middle level. Atoms pile up there, and once more than half the atoms sit in it, emission beats absorption on the middle-to-ground transition. Four-level lasers such as Nd:YAG (1,064 nm) empty the lower laser level quickly too, so a small pump is enough for inversion.
Because every stimulated photon copies the one that triggered it, laser light is coherent: a helium–neon laser’s light can stay in step over tens of centimeters to meters, while a light bulb’s is in step over about a micrometer. That coherence is what makes holograms, fiber-optic interferometers and gravitational-wave detectors possible.
| Source | Frequency / wavelength | Energy per photon | Photons per second | What one photon can do |
|---|---|---|---|---|
| FM station | 100 MHz · 3.0 m | 4.1 × 10⁻⁷ eV | 7.5 × 10²⁹ (50 kW) | Nudge electrons in an antenna |
| Wi-Fi router | 2.4 GHz · 12.5 cm | 9.9 × 10⁻⁶ eV | 6.3 × 10²² (100 mW) | Rock a water molecule |
| Microwave oven | 2.45 GHz · 12.2 cm | 1.0 × 10⁻⁵ eV | 6.2 × 10²⁶ (1 kW) | Same: heat in bulk, no chemistry |
| Thermal glow (you) | 32 THz · 9.4 µm | 0.13 eV | — | Shake a molecular bond |
| TV remote | 319 THz · 940 nm | 1.32 eV | — | Trigger a silicon photodiode |
| Red laser pointer | 461 THz · 650 nm | 1.91 eV | 3.3 × 10¹⁵ (1 mW) | Flip a retinal molecule in your eye |
| UV-C lamp | 1.18 PHz · 254 nm | 4.88 eV | — | Break bonds in DNA |
| Dental X-ray | ~7 EHz · ~40 pm | ~30 keV | — | Ionize thousands of atoms in a cascade |
| Cobalt-60 gamma | 322 EHz · 0.93 pm | 1.33 MeV | — | Same, deeper: radiotherapy |