Overview#

This lesson unifies two classical frameworks: geometric optics (ray picture) and classical mechanics (particle picture). Both obey the same variational principle—Fermat’s principle for light, Hamilton’s principle for particles. The bridge between them is de Broglie’s insight: matter has waves, and the phase of the matter wave is the classical action.

Optical Path as Action#

To avoid symbol overload, we use distinct notation:

• optical path length: \(L = \int n\,\mathrm{d}s\)

• mechanical action: \(S = \int L_{\mathrm{mech}}\,\mathrm{d}t\) (Lagrangian: \(L_{\mathrm{mech}}\))

Then the variational principles are directly parallel:

The physical picture is the same: the observed path is selected by a stationary variational quantity.

De Broglie: Matter Waves#

In 1924, de Broglie proposed that matter, like light, has wave-particle duality.

This unifies particle and wave descriptions. Experimental confirmation (Davisson & Germer, 1927): electrons diffracted by crystals exhibit interference patterns with \(\lambda = h/p\), confirming de Broglie’s hypothesis.

Action = Phase: The Punchline#

In wave optics, the physically observed ray is selected by stationary phase. From §3.1.2,

Just as Fermat’s principle (§3.1.1) governs geometric optics, here we see that the same variational structure appears in mechanics: classical particle trajectories satisfy \(\delta S = 0\). This parallel is no accident; it reveals a deep unity.

For monochromatic light, \(k\) is constant, so

Hence \(\delta\Phi=0 \Leftrightarrow \delta L=0\): the stationary-phase condition is exactly Fermat’s principle. Because this is a variational selection rule, a quantity proportional to \(\Phi\) plays the same role for light as action does in mechanics.

Now compare with mechanics, where the path is selected by

So optics and mechanics have the same structure if we identify an action-like quantity for light with

The proportionality constant is fixed below using matter-wave kinematics. For a free matter wave,

using de Broglie \(\boldsymbol{p}=\hbar\boldsymbol{k}\) and \(E=\hbar\omega\), with classical principal function increment

Therefore,

so the proportionality constant is \(\hbar\), and the universal phase-action relation is \(S=\hbar\Phi\) (up to an additive constant in phase).

So the quantum wavefunction takes the form

This is the unification: action controls phase, and stationary action emerges as the stationary-phase limit.

What is Action?#

Intuitively, action is the scalar functional assigned to a candidate history of a system. Classically, physical histories satisfy a stationary-action condition. Quantum mechanically, each history contributes with phase \(\Phi=S/\hbar\), so action becomes the unifying principle connecting classical dynamics and quantum interference.

The action takes different forms across physics:

Quantization: From Paths to Probability Amplitudes#

Classical mechanics: the observable trajectory satisfies \(\delta S = 0\) (Hamilton’s principle).

Quantum mechanics: the probability amplitude for a particle to travel from \((\boldsymbol{x}_i, t_i)\) to \((\boldsymbol{x}_f, t_f)\) is the propagator \(K(\boldsymbol{x}_f,t_f;\boldsymbol{x}_i,t_i)\), obtained by summing contributions from all possible paths, each weighted by the phase \(\mathrm{e}^{\mathrm{i}S/\hbar}\):

Classical limit (\(S \gg \hbar\)): The phase \(\mathrm{e}^{\mathrm{i}S/\hbar}\) oscillates rapidly. Neighboring paths have wildly different phases and cancel by destructive interference. Only the stationary-phase path—where \(\delta S = 0\)—survives. This is the classical trajectory.

Quantum regime (\(S \sim \hbar\)): All paths have comparable phases and interfere constructively. Diffraction, tunneling, and superposition emerge naturally.

This is Huygens’ principle for matter: every point is a source of “matter wavelets” that propagate and interfere. Quantization is the principle of superposing all these wavelets with the correct quantum phase.

Summary#

• Optics-mechanics duality: Geometric optics and classical mechanics obey identical variational principles. Matter is the “geometric optics” limit of quantum mechanics.

• De Broglie relations: \(\lambda = h/p\) and \(\omega = E/\hbar\) unify waves and particles. Matter-wave phase is \(\Phi = S/\hbar\), so \(\psi \sim \mathrm{e}^{\mathrm{i}S/\hbar}\).

• Path integral quantization: Sum over all paths with phase weight \(\mathrm{e}^{\mathrm{i}S/\hbar}\). The classical path dominates when \(S \gg \hbar\) (stationary phase); quantum interference ruins other paths.

• Actionequals phase: The exponent of the quantum amplitude is the classical action divided by \(\hbar\)—a deep unity connecting classical and quantum mechanics.