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#

• Geometric optics: light follows the path of stationary optical path length: \(L = \int n \, \mathrm{d}s\).

• Classical mechanics: a particle follows the path of stationary action: \(S = \int L \, \mathrm{d}t\).

These have identical structure:

The physical picture: both light and particles are described by a variational principle. The observable path is the one that makes the optical path length (or action) stationary.

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#

Here is the crucial insight connecting classical and quantum mechanics.

In optics, the phase accumulated by a wave traveling through a medium is:

In classical mechanics, the accumulated action along a path is:

De Broglie’s identification: The phase of the quantum wavefunction is the classical action in units of \(\hbar\):

Therefore, the quantum wavefunction is:

This is the bridge. Phase is action. Once we know the action for any physical system, we can immediately write down the phase of its quantum wavefunction.

What is Action?#

The action takes different forms across physics:

In each case, nature chooses the path that makes \(S\) stationary: \(\delta S = 0\).

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 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. Particles are the “geometric optics” of matter.

• De Broglie relations: Matter has wavelength \(\lambda = h/p\) and angular frequency \(\omega = E/\hbar\).

• Action = phase: The quantum wavefunction is \(\psi = A \, \mathrm{e}^{\mathrm{i}S/\hbar}\). This unifies classical and quantum descriptions.

• Path integral quantization: Sum over all paths with phase \(\mathrm{e}^{\mathrm{i}S/\hbar}\). The classical path emerges as the stationary-phase path when \(S \gg \hbar\).

• Classical limit: Stationary phase selects the classical trajectory. Quantum mechanics automatically reproduces classical mechanics in the appropriate limit.