The picture the chapter has built: the world is made of fields, and particles are their quanta. That explains at a stroke why all electrons are exactly alike — each is a quantum of the same field — and why their states are antisymmetric: that is how the field’s operators anticommute. The number of particles is not sacred. Interactions are products of field operators, and the coupling of the Dirac field to the electromagnetic field removes a charged particle, puts one back and adds or removes a photon: emission and absorption are the making and removing of quanta. The square roots of the bosonic operators then have consequences anyone can see. The amplitude for an excited atom to emit into a mode that already holds $n$ photons is $\sqrt{n+1}$ times the amplitude for an empty mode. At $n=0$ the emission still happens: spontaneous emission is emission into the empty modes of the field, which a theory with classical light cannot explain. The extra $n$ is stimulated emission, which makes lasers work. And the vacuum is not nothing: its zero-point fluctuations shift hydrogen’s 2s level — the Lamb shift — and pull plates together.

Ripples on a pond are not things added to the water; they are the water moving. Particles are the ripples of fields, each ripple a whole quantum.

The number of photons $n$ in one mode of an ideal amplifier, every atom excited, against $Gt$. Emitting at the rate $G(n+1)$, spontaneous and stimulated together, the mode fills as $e^{Gt}-1$; with the spontaneous rate alone, dashed, it would fill only as .

An atom with a lower level $\ket{g}$ and an upper level $\ket{e}$, coupled to one mode of light with the strength $g$, has the interaction

• Absorption takes a photon away and raises the atom; emission lowers the atom and adds a photon. The number of photons changes, the number of atoms does not.
• $\bra{g,n+1}\hat{V}\ket{e,n}=g\sqrt{n+1}$: the amplitude to emit into a mode holding photons is times that into an empty mode.

Only the emission term, which lowers the atom and adds a photon, joins $\ket{e,n}$ to $\ket{g,n+1}$. Its matrix element splits into the atom’s part, which is 1, and the photon’s part, which is the square root of the bosonic creation operator. Squared and divided by its value for an empty mode, it gives $n+1$.

The absorption term needs the atom in $\ket{g}$ to start with, and gives zero between these states.

The bosonic creation operator.

Divide by the same quantity for an empty mode.

Emission into the empty mode happens anyway; the rest grows with the light already present.

The amplitude to emit into a mode holding $n$ photons is $\sqrt{n+1}$ times that into an empty mode, so the probability is $n+1$ times larger: 1 spontaneous, stimulated.

Every electron is a quantum of the one electron field, so no property can tell two electrons apart.

An excited atom in empty space still emits: the $n+1$ has its 1 even when $n=0$. A theory with classical light and no light present predicts no emission.

The interaction keeps the number of atoms and changes the number of photons by one: absorption takes one away, emission adds one.

Great work! You now know how to:

• describe particles as the quanta of fields
• derive the $n+1$ of emission and split it into spontaneous and stimulated
• say what the vacuum’s fluctuations do