Photon properties — E = hf, p = h - λ: Visual walkthrough

We are hunting one boxed result: $p = \frac{h}{λ} .$ But we won't write it down and trust it — we'll earn it.

Step 1 — What "momentum" even means (the shove)

WHAT. Before any formula, picture a moving thing hitting a wall. Momentum is the "amount of shove stored in motion" — the thing that transfers to whatever you hit.

WHY start here. People memorise $p = m v$ ("mass times velocity") and think momentum is mass. It isn't. Momentum is a shove budget: how hard a moving object pushes what it strikes. A truck at walking pace and a bullet both deliver a big shove for different reasons. Keeping this picture — shove, not mass — is the whole key to letting light have momentum later.

PICTURE. Two objects hit a wall. The length of the red arrow = how much the wall recoils. That red arrow is $p$ .

Figure — Photon properties — E = hf, p = h - λ

Step 2 — Why the school formula $p = m v$ breaks for light

WHAT. Write the familiar rule and test it on a photon.

$p = mass m velocity v$

Each symbol: $m$ = how much stuff, $v$ = how fast. Multiply them, get the shove.

WHY it fails. A photon (a lump of light) has zero rest mass: $m_{0} = 0$ . Plug that in: $p = 0 \times c = 0.$ This predicts light carries no shove. But comet tails point away from the Sun, and solar sails get pushed — light does shove things (see Radiation Pressure). So $p = m v$ must be an incomplete rule. We need the real one.

PICTURE. The formula $p = m v$ is a low-speed ramp; light lives at the top speed $c$ where the ramp no longer describes reality. The red "actual" curve stays above zero even where the black $m v$ line would send a massless thing to nothing.

Step 3 — The real rule: the energy–momentum triangle

WHAT. Relativity replaces $p = m v$ with a relation between three quantities — total energy $E$ , momentum $p$ , and rest mass $m_{0}$ (see Relativistic Energy-Momentum Relation):

$E^{2} = motion part (p c)^{2} + rest-mass part (m_{0} c^{2})^{2}$

Term by term:

$E$ = the object's total energy.
$p c$ = the piece of that energy stored in motion ( $c$ = speed of light, $3 \times 1 0^{8}$ m/s, just a unit-fixing constant here).
$m_{0} c^{2}$ = the piece of energy locked up in just existing (rest energy).

WHY this tool. We need something that connects momentum to energy without forcing us to multiply by mass. Notice this equation has the exact shape of the Pythagorean theorem $hyp^{2} = leg^{2} + leg^{2}$ . That's not a coincidence — energy, momentum-energy, and rest-energy form a right triangle. Pythagoras is the tool because these three quantities combine like perpendicular sides.

PICTURE. A right triangle: horizontal leg $p c$ (motion), vertical leg $m_{0} c^{2}$ (rest), hypotenuse $E$ (total). The red hypotenuse is what we can measure.

Step 4 — Collapse the triangle: set the rest mass to zero

WHAT. Put $m_{0} = 0$ into the triangle.

$E^{2} = (p c)^{2} + = 0 (0 \cdot c^{2})^{2} = (p c)^{2}$

Take the positive square root (energy is positive): $E = p c ⟹ p = \frac{E}{c} .$

Each symbol now: $E$ = photon's energy, $c$ = light speed, $p$ = its momentum.

WHY this is the crucial move. Killing the vertical leg $m_{0} c^{2}$ flattens the triangle onto its base. The hypotenuse and the horizontal leg become the same line: $E = p c$ . So for light, all its energy is motion-energy — there's nothing "at rest" about it. This is why a massless thing still has momentum: $p = E / c$ , and its energy $E$ is not zero.

PICTURE. The triangle from Step 3, but the top vertex slides down to the base — the vertical leg shrinks to nothing, hypotenuse (red) lies flat on the horizontal leg.

Step 5 — Feed in the photon's energy $E = h f$

WHAT. We already know each photon's energy is a fixed packet set by its frequency (built in the parent from Planck's Law and Blackbody Radiation): $E = Planck’s constant h frequency f$

$h = 6.626 \times 1 0^{- 34}$ J·s — the size of one energy "chunk" per unit frequency.
$f$ = how many wave cycles pass per second (the "colour" of the light).

Substitute this $E$ into $p = E / c$ : $p = \frac{E}{c} = \frac{h f}{c} .$

WHY. $p = E / c$ is true for any massless thing. To make it about this photon we replace its energy with the specific value $h f$ . Nothing new assumed — just plugging one earned result into another.

PICTURE. A conveyor: the "colour dial" $f$ feeds the energy machine $E = h f$ , whose output pours into the momentum machine $p = E / c$ . Red highlights the final momentum output.

Step 6 — Trade frequency for wavelength using $c = f λ$

WHAT. A wave's speed equals its frequency times its wavelength: $c = cycles/sec f \times length/cycle λ$

$f$ = cycles per second, $λ$ = length of one cycle (metres per cycle).
Multiply them: (cycles/sec) $\times$ (metres/cycle) = metres/sec = a speed, and for light that speed is always $c$ .

Rearrange to isolate the ratio we need: $c = f λ ⟹ \frac{f}{c} = \frac{1}{λ} .$

Now take our momentum from Step 5 and swap $f / c$ for $1/ λ$ : $p = \frac{h f}{c} = h \cdot \frac{f}{c} = h \cdot \frac{1}{λ} .$

WHY do this swap. In experiments we usually measure wavelength $λ$ (with a diffraction grating), not raw frequency. Rewriting in $λ$ makes the formula directly usable. The bridge $c = f λ$ is what lets frequency-language and wavelength-language talk.

PICTURE. One wave. Its horizontal repeat-length is labelled $λ$ (red); it slides right at speed $c$ ; the number of crests passing the fixed dot per second is $f$ . The three quantities lock together as $c = f λ$ .

Step 7 — The result, and what it says about every wavelength

WHAT. Collect the chain: $p = \frac{h}{λ}$

$h$ = fixed Planck constant (the "chunk size").
$λ$ = the photon's wavelength.
$p$ = its momentum.

WHY it's believable. Read the shape: $p \propto 1/ λ$ . Short wavelength $\to$ big momentum; long wavelength $\to$ small momentum. Check the extremes (the "all cases" the contract demands):

Case	$λ$	$p = h / λ$	Meaning
Gamma ray	tiny	huge	hits like a hammer (Compton kicks electrons — see Compton Effect)
Visible light	~500 nm	tiny	pushes solar sails only faintly
Radio wave	huge	~0	almost no shove
$λ \to \infty$	infinite	$\to 0$	no wave, no push — consistent
$λ \to 0$	zero	$\to \infty$	ever-shorter light = ever-harder punch

No wavelength gives a nonsensical answer — the formula behaves everywhere.

PICTURE. Three photons drawn as waves — short (red, big arrow), medium, long (tiny arrow). Arrow length = momentum, shrinking as $λ$ grows.

The one-picture summary

Every step, compressed: colour dial $f$ → energy $E = h f$ → collapse the relativity triangle ( $m_{0} = 0$ ) into $E = p c$ → divide by $c$ → swap $f / c$ for $1/ λ$ → land on $p = h / λ$ .

Recall Feynman: the whole walk in plain words

Momentum is just shove — how hard a moving thing pushes what it hits. School says shove = mass × speed, but that's only the slow-lane rule. The real rule of the universe is a right triangle: total energy is the long side, "motion-energy" and "rest-energy" are the two short sides. A photon is pure light — it has no rest side at all, so its triangle flattens into a straight line and total energy equals motion-energy: $E = p c$ . Divide by $c$ and the photon's momentum is just its energy shared out by the speed of light. Its energy is set by colour ( $E = h f$ ), and colour and wavelength are two names for the same thing ( $c = f λ$ ), so in the end the shove is simply Planck's chunk divided by the wavelength: $p = h / λ$ . Bluer (shorter $λ$ ) = harder punch; redder (longer $λ$ ) = gentler nudge; radio waves barely tickle. Nothing weighed a thing — yet the wall still moves.

Connections

Relativistic Energy-Momentum Relation — the triangle we flattened in Steps 3–4.
Compton Effect — experiment where $p = h / λ$ literally kicks electrons.
Radiation Pressure — the macroscopic shove of Step 1.
de Broglie Wavelength — runs $p = h / λ$ backwards for matter.
Wave-Particle Duality — why one lump is both wave ( $λ$ ) and particle ( $p$ ).
Photoelectric Effect · Planck's Law and Blackbody Radiation — where $E = h f$ came from.