Hamilton's principle/Applications/The optical-mechanical analogy

Plain-language version: Light and particles

The optical-mechanical analogy is the correspondence between the paths of light rays and the trajectories of particles. It is not a loose resemblance: the two are governed by the same equation under a single substitution, and the correspondence was Hamilton's route into mechanics rather than a consequence he drew from it. Ninety-two years later it was the route into quantum mechanics.

The correspondence

Wavefronts and the rays orthogonal to them.

Two results already derived in this volume can be set side by side.

Light, by Fermat's principle on Fermat's principle, makes stationary

$\int n(q)ds\mathrm{(}\mathrm{1}\mathrm{)}$

A particle of energy $E$ in a potential $V$, by Jacobi's form of Maupertuis' principle on Abbreviated action: Maupertuis versus Hamilton, makes stationary

$\int \sqrt{2m(E-V(q))}ds\mathrm{(}\mathrm{2}\mathrm{)}$

These are the same integral under

$n(q)⟷\sqrt{2m(E-V(q))}\mathrm{(}\mathrm{3}\mathrm{)}$

Every consequence therefore transfers. The dictionary:

optics	mechanics
refractive index $n(q)$	$\sqrt{2m(E-V)}$
optical path length $\int nds$	abbreviated action $W$
ray	trajectory
wavefront, $\Psi =\text{const}$	surface of constant $W$
Snell's law $n\sin \theta =\text{const}$	conservation of transverse momentum
focus	conjugate point
caustic	envelope of the trajectory family
eikonal equation	Hamilton-Jacobi equation
geometrical optics	classical mechanics
wave optics	(vacant until 1926)

The vacancy in the last row is the point of this page.

Both are Hamilton-Jacobi

The optical-mechanical dictionary.

Write the eikonal equation of geometrical optics and the time-independent Hamilton-Jacobi equation of the Hamilton-Jacobi equation together:

$|\mathrm{\nabla }\Psi {|}^2=n^2(q),|\mathrm{\nabla }W{|}^2=2m(E-V(q))\mathrm{(}\mathrm{4}\mathrm{)}$

One equation, two readings, under (3). Rays are orthogonal to surfaces of constant $\Psi$; trajectories are orthogonal to surfaces of constant $W$. The orthogonality follows from $p=\mathrm{\nabla }W$, quoted from Hamilton on Action as a function of its endpoints as his equation (C.).

Hamilton's own order of discovery

The empty corner of the optical-mechanical analogy.

Hamilton wrote the optics first. His Theory of Systems of Rays introduced the characteristic function for optical systems, and the dynamics of 1834 transposed it. He says as much in the introduction to the essay: the dynamical principle is another form of an idea already applied to optics in that theory, and he had announced the intention of applying it to the motion of systems of bodies when the optical theory was published.

He adds a remark whose reach he could not have measured: that the manner of calculation exemplified in optics and dynamics seems not confined to those two sciences, and may constitute a separate branch of analysis when matured by the future labours of mathematicians.

That is a claim about method, made in 1834, and it was vindicated by a physics he had no reason to anticipate.

What the analogy is missing

The correspondence is between geometrical optics and classical mechanics. Geometrical optics is not the whole of optics: it is the limit of wave optics when the wavelength is negligible against the scale of the apparatus. Rays are what waves look like when diffraction is invisible.

So the table has a hole. If mechanics corresponds to geometrical optics, and geometrical optics is an approximation to something, then classical mechanics should be an approximation to something too — a wave mechanics, of which it is the short-wavelength limit.

Nothing in the nineteenth century supplied a reason to take that seriously. There was no evidence for a wavelength, no phenomenon demanding one, and no candidate for what would be waving. The analogy sat complete and unexploited for ninety-two years.

Filling the hole

Assume the missing wave and see what it must satisfy. Write

$\psi =A(q)\exp \left(\frac{iW(q)}{\hslash }\right)\mathrm{(}\mathrm{5}\mathrm{)}$

with $\hslash$ a constant of the dimensions of action, fixing the wavelength scale. Substitute into the wave equation ${\mathrm{\nabla }}^2\psi +k^2\psi =0$ and separate real and imaginary parts. The real part is

$|\mathrm{\nabla }W{|}^2-{\hslash }^2\frac{{\mathrm{\nabla }}^{2}A}{A}={\hslash }^2{k}^2\mathrm{(}\mathrm{6}\mathrm{)}$

As $\hslash \to 0$ the second term drops and (6) becomes the second equation of (4). Classical mechanics is the $\hslash \to 0$ limit of a wave equation, and the Hamilton-Jacobi equation is its eikonal.

Running the argument backwards — demanding a wave equation whose short-wavelength limit is Hamilton-Jacobi, with $\hslash$ not zero — produces

$-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\psi +V\psi =E\psi \mathrm{(}\mathrm{7}\mathrm{)}$

the time-independent Schrödinger equation. That is how Schrödinger constructed it in 1926, and the derivation is the analogy read in the direction Hamilton could not read it.

The retained term in (6), $-{\hslash }^2{\mathrm{\nabla }}^{2}A/A$, is the correction that classical mechanics discards. It is what makes particles diffract.

Provenance: the account of Schrödinger's 1926 construction is editorial summary. No primary source for it has been verified for this volume, and equation (7) is standard rather than transcribed. The Hamilton citations above are ESTABLISHED. See Sources.

Why $\hslash$ had to be small

The analogy explains why classical mechanics works. The wavelength associated with a particle by (5) is $\lambda =2\pi \hslash /p$, and for any laboratory-scale object it is smaller than any length in the apparatus by many orders of magnitude. Geometrical optics is exact in the same limit.

Where the wavelength is comparable to the scale of variation of the potential, the ray picture fails and the wave picture is required. Electron diffraction is that regime, and it stands to classical mechanics exactly as diffraction through a narrow slit stands to ray optics.

Foci, conjugate points, and phases

The analogy makes one of this volume's recurring themes concrete. A focus in optics, a conjugate point in the variational theory of the Jacobi condition and conjugate points, and a caustic of a trajectory family are the same object under (3).

At such points $W$ is multivalued, the eikonal approximation breaks down, and the amplitude $A$ in (5) diverges. Wave optics resolves the divergence and leaves behind a phase shift each time a ray passes a focus. The same shift appears in mechanics as the phase correction counting conjugate points crossed, and it reappears in the path integral on the Feynman path integral.

One phenomenon, four vocabularies. That is the analogy's most useful service: a result proved in any one of the four transfers to the others without reproof.

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Central force and the Kepler problem works the case Hamilton himself worked, in his own notation.