From Light Quanta to Photons

The route from light quanta to photons is not a single discovery moment. Einstein’s light quantum was a hypothesis about discrete radiation energy exchange. A photon is one excitation of the quantized electromagnetic field, with energy, momentum, polarization, quantum statistics, and no classical trajectory in general.

A useful distinction is between the field, which admits many different quantum states, and one photon, which is a single excitation. A state of light may contain exactly one photon, many photons, or a superposition of photon numbers.

Light Quantum Language

Einstein’s 1905 proposal used light quanta to explain why certain radiation processes behave as if energy is exchanged in packets of size

$$E=h\nu.$$

That was already radical. It applied discreteness to radiation itself in contexts such as the photoelectric effect, not merely to material oscillators in a blackbody calculation.

But the early light quantum was not yet the fully modern photon. Several features were missing or unsettled:

• a coherent account of interference and diffraction with single quanta;
• a mature description of photon momentum and recoil;
• a quantum theory of emission and absorption;
• polarization as part of a quantum state;
• multiphoton states and photon statistics;
• a field-theoretic account in which photon number can change.

Photon Terminology

The word “photon” entered the literature in the 1920s, after the photoelectric and Compton evidence had made light quanta difficult to avoid. Gilbert N. Lewis introduced the term in 1926, although his proposed conservation of photons was not the modern quantum-electrodynamic account: atoms routinely emit and absorb photons.

The name survived because it was useful. It gave physicists a compact term for the quantum of electromagnetic radiation, especially once quantum mechanics and radiation theory began to supply better mathematical machinery.

Terminology should not hide the conceptual evolution:

Term	Historical role	Modern caution
Planck energy element	Energy scale in oscillator counting	Not yet a photon
Einstein light quantum	Radiation energy packet in exchange processes	Not yet full QED
Photon	Quantum of the electromagnetic field	Not a tiny classical particle

This is why saying “Planck discovered photons” is too compressed. Planck made $h$ unavoidable in radiation theory. Einstein made light-quanta energy exchange explicit. Compton strengthened momentum evidence. Quantum theory and field quantization gave the modern photon its home.

Why Photons Are Not Little Classical Bullets

Photons can arrive as localized detector events, and in scattering they carry energy and momentum. Those facts tempt a classical particle picture. But photons are not little bullets for several reasons.

First, a single-photon state can produce interference. The observed pattern comes from probability amplitudes, not from a small object secretly choosing one classical path while ignoring the other.

Second, photon localization is subtle. Photons are massless spin-1 quanta with gauge constraints and no rest frame. Position-space language for photons is not as straightforward as position-space wavefunctions for massive nonrelativistic particles.

Third, the same electromagnetic field admits states with very different photon-number properties: number states, coherent states, thermal states, squeezed states, and entangled states. A classical ray picture captures only a limiting regime.

The safer statement is:

$$E=\hbar\omega, \qquad \mathbf p=\hbar\mathbf k$$

for a photon mode in vacuum, while the state itself belongs to quantum theory.

Photon States in Quantum Optics

Quantum optics treats light as a quantum system. A field mode behaves like a harmonic oscillator, and a number state contains a definite occupation number:

$$\lvert n\rangle \quad \text{has photon number } n.$$

For a mode labeled by wavevector $\mathbf k$ and polarization $s$, a one-photon number state is often written schematically as

$$\lvert 1_{\mathbf k,s}\rangle.$$

A realistic single photon is usually a wavepacket, a superposition of many modes:

$$\lvert 1_f\rangle = \sum_s \int d^3k\, f_s(\mathbf k) \lvert 1_{\mathbf k,s}\rangle,$$

Here choose $\langle1_{\mathbf k,s}|1_{\mathbf k\prime,s\prime}\rangle=\delta_{ss\prime}\delta^{(3)}(\mathbf k-\mathbf k\prime)$. The exact plane-wave kets are generalized states, and the normalizable packet satisfies

$$\sum_{s=1}^{2}\int d^3k\,|f_s(\mathbf k)|^2=1.$$

The two labels $s$ describe the transverse polarizations. The coefficients $f_s$ give the photon’s spectral, directional, and polarization content. Other continuum normalizations move factors between the measure, the coefficients, and the kets.

Not all important light states have definite photon number. A coherent state, for example, has a Poissonian photon-number distribution and often approximates a classical laser field. That is why quantum optics does not reduce to “counting particles”; it studies the quantum states of field modes.

Photons in Quantum Field Theory

In quantum field theory, the electromagnetic field is quantized. Each free field mode acts like an oscillator with creation and annihilation operators. Schematic one-photon states are created from the vacuum by

$$\lvert 1_{\mathbf k,s}\rangle = a^\dagger_{\mathbf k,s} \lvert 0\rangle.$$

The Hamiltonian and momentum operators count mode occupations, with each photon contributing

$$E=\hbar\omega, \qquad \mathbf p=\hbar\mathbf k.$$

This field-theoretic view explains why photon number is not generally conserved. Atoms can emit or absorb photons. Charged particles can radiate. Photon pairs can be created in nonlinear or high-energy processes when the full interaction permits it. The conserved quantities are total energy, momentum, angular momentum, charge, and other symmetries, not photon number in general.

For the operator construction, continue with Harmonic Oscillator to Fields, Fock Space, and Second Quantization.

From Measured Relations to Field States

The path can be summarized as a sequence of increasingly precise claims:

Step	What it contributed
Blackbody radiation	Introduced a universal quantum scale in radiation equilibrium
Einstein light quantum	Proposed discrete radiation energy exchange $h\nu$
Photoelectric effect	Supported frequency-controlled energy transfer
Millikan measurements	Confirmed the photoelectric slope with high precision
Compton scattering	Supported photon momentum and recoil kinematics
Early photon debates	Tested alternatives and clarified what evidence did not yet prove
Quantum optics and QED	Made photons field quanta with states, statistics, and interactions

The point is cumulative. No single experiment supplied the whole modern photon. The photon concept became trustworthy because several independent strands converged and because later theory explained how the apparently contradictory wave and particle features fit into one framework.

Common Mistakes

• Treating “light quantum” and “photon” as identical in every historical context.
• Saying a photon is simply a tiny localized pellet of light.
• Thinking single-photon interference is a contradiction. It is a feature of quantum amplitudes.
• Forgetting polarization and mode structure.
• Assuming photon number is always conserved.
• Importing full QED conclusions into Planck’s or Einstein’s earliest papers.

Exercises

1. Give two reasons why Einstein’s light quantum should not be identified too quickly with the modern photon.

Solution

Einstein’s light quantum captured discrete energy exchange $h\nu$, especially in contexts such as the photoelectric effect. It did not yet provide a full field-theoretic account of radiation, photon-number states, polarization, quantum statistics, or emission and absorption dynamics. Those features required later quantum mechanics, quantum optics, and quantum electrodynamics.

2. Explain why localized photon detection events do not imply that photons are classical bullets.

Solution

A detector event can be localized even when the quantum state leading to it is spatially extended or in a superposition of modes. Single-photon interference shows that amplitudes, not classical trajectories, govern the probabilities. The localized event is an interaction outcome, not evidence that the photon followed a tiny classical path beforehand.

3. In two orthogonal modes, consider $|\psi\rangle=(|1,0\rangle+e^{i\phi}|0,1\rangle)/\sqrt2$. Is its total photon number uncertain? What happens if the two modes are coherently recombined?

Solution

Both terms have total photon number one, so $\hat N|\psi\rangle=|\psi\rangle$: the number is definite even though the occupied mode is not. A balanced recombination into the sum and difference modes gives probabilities $(1+\cos\phi)/2$ and $(1-\cos\phi)/2$. One photon can therefore have a definite number and phase-dependent interference between mode alternatives.

References

• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley, 1989.
• P. A. M. Dirac, “The Quantum Theory of the Emission and Absorption of Radiation,” Proceedings of the Royal Society A 114, 243-265 (1927), DOI: 10.1098/rspa.1927.0039.
• A. Einstein, “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt,” Annalen der Physik 322, 132-148 (1905), DOI: 10.1002/andp.19053220607.
• R. J. Glauber, “Coherent and Incoherent States of the Radiation Field,” Physical Review 131, 2766-2788 (1963), DOI: 10.1103/PhysRev.131.2766.
• G. N. Lewis, “The Conservation of Photons,” Nature 118, 874–875 (1926), DOI: 10.1038/118874a0.
• R. Loudon, The Quantum Theory of Light, 3rd ed., Oxford University Press, 2000.