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

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

	Light, by ~~Fermat's principle on~~		Light, by [[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]], makes stationary
	[[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]], makes stationary

	:<math>\int n(q)\; ds		:<math>\int n(q)\; ds

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	Light, by Fermat's principle on		Light, by Fermat's principle on
	[[Hamilton's principle/Applications/Fermat's principle]], makes stationary		[[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]], makes stationary

	:<math>\int n(q)\; ds		:<math>\int n(q)\; ds
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	A particle of energy <math>E</math> in a potential <math>V</math>, by Jacobi's		A particle of energy <math>E</math> in a potential <math>V</math>, by Jacobi's
	form of Maupertuis' principle on		form of Maupertuis' principle on
	[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]],		[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]],
	makes stationary		makes stationary

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	Write the eikonal equation of geometrical optics and the time-independent		Write the eikonal equation of geometrical optics and the time-independent
	Hamilton-Jacobi equation of		Hamilton-Jacobi equation of
	[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] together:		[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]] together:

	:<math>\vert \nabla\Psi \vert^2 = n^2(q),		:<math>\vert \nabla\Psi \vert^2 = n^2(q),
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	<math>W</math>. The orthogonality follows from		<math>W</math>. The orthogonality follows from
	<math>p = \nabla W</math>, quoted from Hamilton on		<math>p = \nabla W</math>, quoted from Hamilton on
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] as his		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]] as his
	equation (C.).		equation (C.).

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	No primary source for it has been verified for this volume, and equation (7) is		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.		standard rather than transcribed. The Hamilton citations above are ESTABLISHED.
	See [[Hamilton's principle/Sources]].		See [[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

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

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	same shift appears in mechanics as the phase correction counting conjugate points		same shift appears in mechanics as the phase correction counting conjugate points
	crossed, and it reappears in the path integral on		crossed, and it reappears in the path integral on
	[[Hamilton's principle/Applications/The Feynman path integral]].		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]].

	'''One phenomenon, four vocabularies.''' That is the analogy's most useful		'''One phenomenon, four vocabularies.''' That is the analogy's most useful
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	== Next ==		== Next ==

	[[Hamilton's principle/Applications/Central force and the Kepler problem]] works		[[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]] works
	the case Hamilton himself worked, in his own notation.		the case Hamilton himself worked, in his own notation.

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			{{DISPLAYTITLE:The optical-mechanical analogy}}
	{{Plebeian\|Hamilton's principle/Explainers/Light and particles}}		{{Plebeian\|Hamilton's principle/Explainers/Light and particles}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/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 ==

[[File:OPL-HP-F-12.png|thumb|right|400px|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
[[Hamilton's principle/Applications/Fermat's principle]], makes stationary

:<math>\int n(q)\; ds
\qquad\qquad\mathrm{(1)}</math>

A particle of energy <math>E</math> in a potential <math>V</math>, by Jacobi's
form of Maupertuis' principle on
[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]],
makes stationary

:<math>\int \sqrt{2m\bigl(E - V(q)\bigr)}\; ds
\qquad\qquad\mathrm{(2)}</math>

These are the same integral under

:<math>n(q) \;\longleftrightarrow\; \sqrt{2m\bigl(E - V(q)\bigr)}
\qquad\qquad\mathrm{(3)}</math>

Every consequence therefore transfers. The dictionary:

{| class="wikitable"
! optics !! mechanics
|-
| refractive index <math>n(q)</math> || <math>\sqrt{2m(E-V)}</math>
|-
| optical path length <math>\int n\,ds</math> || abbreviated action <math>W</math>
|-
| ray || trajectory
|-
| wavefront, <math>\Psi = \text{const}</math> || surface of constant <math>W</math>
|-
| Snell's law <math>n\sin\theta = \text{const}</math> || 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 ==

[[File:OPL-HP-F-38.png|thumb|right|400px|The optical-mechanical dictionary.]]

Write the eikonal equation of geometrical optics and the time-independent
Hamilton-Jacobi equation of
[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] together:

:<math>\vert \nabla\Psi \vert^2 = n^2(q),
\qquad
\vert \nabla W \vert^2 = 2m\bigl(E - V(q)\bigr)
\qquad\qquad\mathrm{(4)}</math>

One equation, two readings, under (3). Rays are orthogonal to surfaces of
constant <math>\Psi</math>; trajectories are orthogonal to surfaces of constant
<math>W</math>. The orthogonality follows from
<math>p = \nabla W</math>, quoted from Hamilton on
[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] as his
equation (C.).

== Hamilton's own order of discovery ==

[[File:OPL-HP-F-48.png|thumb|right|400px|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.<ref>Hamilton, ''On a General Method in
Dynamics'' (1834), Introductory Remarks. Verbatim in substance. ESTABLISHED
source.</ref>

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.<ref>Hamilton, ''On a General Method in
Dynamics'' (1834), Introductory Remarks. Verbatim in substance. ESTABLISHED
source.</ref>

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

:<math>\psi = A(q)\, \exp\!\left( \frac{i\,W(q)}{\hbar} \right)
\qquad\qquad\mathrm{(5)}</math>

with <math>\hbar</math> a constant of the dimensions of action, fixing the
wavelength scale. Substitute into the wave equation
<math>\nabla^2\psi + k^2\psi = 0</math> and separate real and imaginary parts. The
real part is

:<math>\vert \nabla W \vert^2 - \hbar^2\,\frac{\nabla^2 A}{A} = \hbar^2 k^2
\qquad\qquad\mathrm{(6)}</math>

As <math>\hbar \to 0</math> the second term drops and (6) becomes the second
equation of (4). '''Classical mechanics is the <math>\hbar \to 0</math> 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 <math>\hbar</math> not zero — produces

:<math>-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi
\qquad\qquad\mathrm{(7)}</math>

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), <math>-\hbar^2\nabla^2 A/A</math>, is the correction
that classical mechanics discards. It is what makes particles diffract.

<blockquote>
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 [[Hamilton's principle/Sources]].
</blockquote>

== Why <math>\hbar</math> had to be small ==

The analogy explains why classical mechanics works. The wavelength associated
with a particle by (5) is <math>\lambda = 2\pi\hbar/p</math>, 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
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]],
and a '''caustic''' of a trajectory family are the same object under (3).

At such points <math>W</math> is multivalued, the eikonal approximation breaks
down, and the amplitude <math>A</math> 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
[[Hamilton's principle/Applications/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.

== Next ==

[[Hamilton's principle/Applications/Central force and the Kepler problem]] works
the case Hamilton himself worked, in his own notation.

== References ==

<references />