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

Before you can read $E=hf$ or $p=h/\lambda$, you need to know what a wave is, what "frequency" and "wavelength" mean as pictures, what the speed $c$ ties them together with, and what "momentum" and "rest mass" really are. We build them one at a time — each new symbol earns its place before the next arrives.

1. A wave, as a picture

Look at the top curve in the figure below. It is a snapshot — one frozen instant. The curve rises to a crest, dips to a trough, rises again. That repeating shape is the only thing you need to define everything else on this page.

• The height of a crest is called the amplitude — how tall the wiggle is. In everyday terms, bigger amplitude = brighter light.
• The horizontal distance from one crest to the very next crest is called the wavelength.

We will need both, but for photons the surprise (built in the parent note) is that amplitude turns out not to set a single photon's energy — wavelength does.

2. Wavelength — the symbol $\lambda$

Why do we need it? Because "color" for light is wavelength. Red light has a long $\lambda$ (about $700$ nm), blue light a short $\lambda$ (about $450$ nm). When the parent note writes $p=h/\lambda$, it is saying momentum is tied to this crest-to-crest distance.

3. Frequency — the symbol $f$

Why do we need a second description of the same wave? Because $\lambda$ is a picture in space (a snapshot), while $f$ is a picture in time (stand still and count crests going by). Both describe the same wave from different viewpoints.

In the figure, the cyan dot sits at a fixed spot while the wave streams past it. Every time a crest reaches the dot, we tick a counter. The number of ticks in one second is $f$. Fast wiggling (crests close together, small $\lambda$) → many ticks per second → large $f$. This is your first clue that $\lambda$ and $f$ are inversely linked — the next section makes that exact.

4. The bridge — speed $c$ and $c=f\lambda$

Now the key relation that ties the ruler ($\lambda$) to the clock ($f$):

WHY is this true? In one second, $f$ crests pass you (that's what $f$ means). Each crest is $\lambda$ metres long. So the total length of wave that swept past in one second is (number of crests) $\times$ (length of each) $=f\times \lambda$. But "length passing per second" is exactly speed. Hence $c=f\lambda$.

5. Planck's constant $h$ — the exchange rate

Why does the topic need it? $f$ is measured in cycles-per-second; energy is measured in joules. To turn one into the other you need a conversion factor with the right units. That factor is $h$. In $E=hf$, $h$ is literally the price tag: each hertz of frequency costs one $h$ of energy.

6. Energy $E$ and the electronvolt

Why introduce a second energy unit? A single photon carries something like $10^{-19}\text{J}$ — a horrible number to read. Divide by $1.6\times 10^{-19}$ and you get a friendly few "electronvolts." A green photon is about $2.5\text{eV}$; that's a number a human can hold in mind.

7. Momentum $p$ — before relativity

We meet $p$ here so that the parent note's $p=h/\lambda$ has a home. But there's a trap: the everyday formula $p=mv$ says a massless thing ($m=0$) has zero momentum. Light has zero rest mass yet clearly pushes things (comet tails, solar sails). So $p=mv$ cannot be the whole story — we need the deeper version, coming next.

8. Rest mass $m_0$ and the relativistic upgrade

Einstein's relativity replaces the schoolbook $p=mv$ with a fuller law connecting energy, momentum, and rest mass:

You don't derive this here (it comes from Relativistic Energy-Momentum Relation) — you just need to read it. It says: a thing's energy is built from two pieces, a motion piece $(pc)$ and a rest piece $(m_0{c}^2)$, combined like the two legs of a right triangle.

In the figure the hypotenuse is $E$, one leg is $pc$, the other leg is $m_0{c}^2$. For a photon the rest-mass leg shrinks to zero, so the triangle collapses flat and the hypotenuse equals the remaining leg: $m_0=0⟹E^2=(pc{)}^2⟹E=pc⟹p=\frac{E}{c}.$

That single collapse is the engine behind $p=h/\lambda$ on the parent note. Now $p=E/c$ gives a non-zero momentum even with zero mass — the everyday formula's trap is dodged.

How every symbol feeds the two formulas

Read it top-to-bottom: pictures of a wave give you $\lambda$ and $f$; the speed $c$ bridges them; $h$ turns $f$ into energy $E$; the relativistic triangle with $m_0=0$ turns energy into momentum $p$. Every arrow is a step this page justified.

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Parent topic — Photon properties — where these symbols combine into $E=hf$ and $p=h/\lambda$.
• Relativistic Energy-Momentum Relation — full justification of the energy–momentum triangle used in §8.
• Planck's Law and Blackbody Radiation — where $h$ and the idea of energy chunks were born.
• Photoelectric Effect — the experiment that forced "energy depends on $f$, not amplitude."
• de Broglie Wavelength — reuses $p=h/\lambda$ for matter.
• Wave-Particle Duality — the big picture these foundations serve.