Photon Momentum (p = h/λ)

Light pushes on what it strikes, and it does so without having any mass. That sounds like a contradiction only if you believe momentum is defined as $mv$. It is not; $mv$ is the low-speed approximation. The general relation in relativity is $E^2 = (pc)^2 + (mc^2)^2$, and setting $m = 0$ leaves $E = pc$. Divide the photon energy $E = hc/\lambda$ by $c$ and you have $p = h/\lambda$. So photon momentum is not an extra assumption bolted on to quantum theory — it is what you get by combining the energy of a photon with the massless case of relativity, and it could hardly have come out any other way.

A green photon at 500 nm carries $6.626 \times 10^{-34}/500 \times 10^{-9} = 1.33 \times 10^{-27}$ kg·m/s. Individually negligible, collectively not. At Earth's distance the Sun delivers about 1361 W/m², and dividing by $c$ gives a radiation pressure of $4.5$ µPa on a surface that absorbs the light — roughly 4.5 µN per square metre. A perfect mirror reverses each photon rather than stopping it, so the momentum change doubles and the pressure becomes about 9 µPa. A sail of one square kilometre therefore feels around 9 N: less than the weight of a litre of water, but continuous, free, and needing no propellant.

This is one of the rare places where the classical and quantum accounts agree on the number and disagree only on the picture. Maxwell predicted radiation pressure from electromagnetism in 1873, decades before anyone spoke of photons, and Pyotr Lebedev measured it with a torsion balance in 1899–1901. The photon reading was tested directly by Arthur Compton in 1923: X-rays scattering from electrons come away with a longer wavelength, shifted by exactly the amount that conservation of energy and momentum demands if the incoming X-ray is a particle carrying $p = h/\lambda$. That experiment, more than the photoelectric effect, is what persuaded the profession that light quanta were real. De Broglie inverted the same equation the following year.

Three corrections worth making. Reflection gives twice the push of absorption, which is why solar sails are aluminized rather than black, and forgetting the factor of two is the usual arithmetic slip. The Crookes radiometer — the little vaned wheel spinning in a bulb in every museum gift shop — is not radiation pressure; it turns the wrong way for that, and it is driven by the residual gas near the warmer black vanes. And the tempting shortcut of assigning the photon a "relativistic mass" $E/c^2$ and writing $p = mc$ gives the right number for the wrong reason: relativistic mass is a discarded convention, and a photon has no rest frame in which any mass could be measured. Momentum here is defined by the energy–momentum relation, and that is the honest account. One more piece of hygiene: comet tails. The dust tail is shaped by radiation pressure, but the separate blue ion tail is blown by the solar wind, which is particles, not light. Saying light pressure makes comet tails is half the story.