Photons as Oscillator Quanta

From oscillators to fields

Throughout this module we have studied the quantum harmonic oscillator in the context of mechanical systems — a particle in a parabolic potential, a vibrating molecule, atoms on a lattice. There is, however, a much deeper application: the electromagnetic field itself is made of harmonic oscillators, one for every mode of the field, and a photon is simply one quantum of excitation in such an oscillator.

This identification is not a metaphor. The same mathematical structure — the same ladder operators, the same equally-spaced energy levels, the same Fock states $|n\rangle$ — governs mechanical vibrations and light. Understanding why requires a brief look at how Maxwell's equations can be reduced to an infinite collection of harmonic oscillators.

Modes of the radiation field

Consider the electromagnetic field inside a cubic cavity of side $L$ with perfectly conducting walls. The boundary conditions force the electric and magnetic fields to form standing waves. Each allowed standing-wave pattern is called a mode, labelled by a wavevector $\mathbf{k}$ and a polarisation index $s$ . The electric field can be written as a sum over modes:

$\mathbf{E}(\mathbf{r},t) = \sum_{\mathbf{k},s} \mathcal{E}_{\mathbf{k}s}(t)\,\mathbf{u}_{\mathbf{k}s}(\mathbf{r}),$

where $\mathbf{u}_{\mathbf{k}s}$ are the spatial mode functions fixed by the boundary conditions and $\mathcal{E}_{\mathbf{k}s}(t)$ are the time-dependent amplitudes.

When Maxwell's equations are written in terms of these amplitudes, each mode decouples from every other mode and the amplitude satisfies

$\ddot{\mathcal{E}}_{\mathbf{k}s} + \omega_k^2\,\mathcal{E}_{\mathbf{k}s} = 0, \qquad \omega_k = c|\mathbf{k}|.$

This is exactly the equation of a simple harmonic oscillator of frequency $\omega_k$ . The total energy stored in the field is the sum of harmonic-oscillator energies, one per mode:

$H = \sum_{\mathbf{k},s} \frac{1}{2}\!\left(\dot{\mathcal{E}}_{\mathbf{k}s}^2 + \omega_k^2\,\mathcal{E}_{\mathbf{k}s}^2\right).$

Quantising the field

Quantising a harmonic oscillator means promoting its coordinate and momentum to operators satisfying the canonical commutation relation $[\hat{x}, \hat{p}] = i\hbar$ . We do exactly the same for each field mode. For mode $(\mathbf{k}, s)$ , introduce the creation and annihilation operators $\hat{a}^\dagger_{\mathbf{k}s}$ and $\hat{a}_{\mathbf{k}s}$ with

$[\hat{a}_{\mathbf{k}s},\, \hat{a}^\dagger_{\mathbf{k}'s'}] = \delta_{\mathbf{k}\mathbf{k}'}\,\delta_{ss'}.$

The Hamiltonian for the entire field then becomes

$\hat{H} = \sum_{\mathbf{k},s} \hbar\omega_k\!\left(\hat{a}^\dagger_{\mathbf{k}s}\hat{a}_{\mathbf{k}s} + \tfrac{1}{2}\right) = \sum_{\mathbf{k},s} \hbar\omega_k\!\left(\hat{N}_{\mathbf{k}s} + \tfrac{1}{2}\right),$

where $\hat{N}_{\mathbf{k}s} = \hat{a}^\dagger_{\mathbf{k}s}\hat{a}_{\mathbf{k}s}$ is the photon number operator for that mode. The eigenvalue $n_{\mathbf{k}s} = 0, 1, 2, \ldots$ gives the number of photons in mode $(\mathbf{k}, s)$ .

The energy eigenvalues are

$E = \sum_{\mathbf{k},s} \hbar\omega_k\!\left(n_{\mathbf{k}s} + \tfrac{1}{2}\right).$

Each photon in mode $(\mathbf{k},s)$ contributes exactly $\hbar\omega_k$ to the total energy. This is Planck's quantisation condition $E = h\nu = \hbar\omega$ recovered as a theorem of quantum mechanics, not a separate postulate.

Photons as quanta of excitation

A single-mode field with $n$ photons occupies the Fock state $|n\rangle$ , defined by $\hat{N}|n\rangle = n|n\rangle$ . The ladder operators act as

$\hat{a}^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle, \qquad \hat{a}|n\rangle = \sqrt{n}\,|n-1\rangle,$

so $\hat{a}^\dagger$ creates one photon and $\hat{a}$ destroys one photon. The vacuum state $|0\rangle$ , defined by $\hat{a}|0\rangle = 0$ , contains no photons but still carries the zero-point energy $\tfrac{1}{2}\hbar\omega$ — the electromagnetic vacuum fluctuation whose physical consequences include the Casimir effect and spontaneous emission.

The state with a definite number of photons is a Fock state; the coherent states familiar from laser optics are superpositions of Fock states with a Poissonian photon-number distribution, and they are the quantum states that most closely mimic classical electromagnetic waves.

Energy levels of a single-mode field

For a single field mode of frequency $\omega$ , the energy spectrum is identical to the mechanical harmonic oscillator:

| Photon number $n$ | Energy $E_n$ | |---|---| | 0 | $\tfrac{1}{2}\hbar\omega$ (vacuum) | | 1 | $\tfrac{3}{2}\hbar\omega$ | | 2 | $\tfrac{5}{2}\hbar\omega$ | | $n$ | $\hbar\omega\!\left(n+\tfrac{1}{2}\right)$ |

The transition energy between adjacent levels is always $\hbar\omega$ : absorbing or emitting one photon shifts the oscillator up or down by exactly one rung of the ladder. This is why atomic emission spectra show sharp lines at frequencies $\nu = (E_i - E_f)/h$ — each spectral line corresponds to the exchange of exactly one photon with a field mode of matching frequency.

Why this matters

The identification of photons with oscillator quanta has profound consequences:

Planck's law follows directly. The average energy of a mode at temperature $T$ is $\langle E \rangle = \hbar\omega / (e^{\hbar\omega/k_B T} - 1) + \tfrac{1}{2}\hbar\omega$ , obtained by computing $\langle n \rangle$ with Bose-Einstein statistics. Summing over all modes reproduces the Planck blackbody spectrum without any arbitrary cut-off.
Quantum optics is harmonic-oscillator physics. Squeezed light, entangled photon pairs, and cat states are all quantum states of field-mode oscillators, prepared and measured with $\hat{a}$ and $\hat{a}^\dagger$ .
Quantum field theory generalises the idea. Every fundamental particle — electrons, quarks, gluons — is a quantum of excitation of its own field oscillator. The quantum harmonic oscillator is not just a pedagogical warm-up; it is the structural template for the entire Standard Model.