Photon Energy (E = hf)

$E = hf$ says that the energy of a single photon depends on the colour of the light and on nothing else. A dim blue lamp and a searchlight of the same blue emit photons of identical energy; the searchlight simply emits far more of them. Brightness counts photons, frequency sizes them. Planck's constant $h = 6.626 \times 10^{-34}$ J·s is the exchange rate between the two descriptions — between how fast the field oscillates and the smallest parcel of energy that oscillation can hand over. The relation is not a definition or a curve fit; it is the statement that light is granular, and that the size of the grain is set by frequency alone.

Take green light at $5.5 \times 10^{14}$ Hz. One photon carries $6.626 \times 10^{-34} \times 5.5 \times 10^{14} = 3.64 \times 10^{-19}$ J. That number is inconveniently small, so divide by the elementary charge $1.602 \times 10^{-19}$ C and read it as 2.27 eV. The electronvolt exists precisely because joules are the wrong size for one photon: a 100 W bulb radiating in the green emits something like $3 \times 10^{20}$ photons every second, which is why the graininess never shows up in ordinary life.

The history is worth getting right, because the popular version flattens it. Planck was not trying to quantize anything in 1900. He was fitting a curve — interpolating between Wien's expression, which worked at short wavelengths, and Rayleigh's, which worked at long ones — and he found in October that a particular formula matched the Berlin measurements across the whole spectrum. Deriving that formula in December forced him to count energy in units of $hf$, a step he later called an act of desperation. Crucially, he attributed the quantization to the oscillators in the cavity wall, not to light itself. It was Einstein in 1905 who took the far stronger position that light is quantized in transit, and that paper — not relativity — won him the Nobel Prize in 1921. Planck resisted it for over a decade; when he recommended Einstein for the Prussian Academy in 1913 he asked that this particular lapse not be held against him.

Three things people get wrong. The first is assuming brighter light means more energetic photons; it does not, and that single point is why ultraviolet at 4 eV damages skin and DNA while red light at 2 eV cannot, at any intensity you care to apply. Photochemistry responds to the energy of the individual photon, and below the threshold nothing happens no matter how much energy you deliver in total. The second is angular frequency: $f$ here is in hertz, and if you are working with $\omega$ in radians per second the constant becomes $\hbar = h/2\pi$. Mixing the two puts the answer out by 6.283. The third is subtle and useful — when light enters glass its frequency does not change; only its speed and wavelength do. The photon's energy is therefore the same in glass as in vacuum, which makes $E = hf$ the safer form to reach for whenever a medium is involved.