Waves and rays
This page explains how mechanics turned into a wave theory. It uses no mathematics. The technical page it goes with is named above.
Looking for the easiest description
the Hamiltonian described motion as a flow in a space of positions and momenta.
Some descriptions make that flow simple and others make it complicated. So it is natural to ask for the best one — a description in which nothing appears to move at all, and every quantity stays put.
Such a description exists. All the difficulty moves into finding it, and the finding is a single equation.
That equation replaces every other equation of motion for the system. Solve it and the entire problem is done, by differentiating rather than by integrating.
This is Hamilton's method, refined by Jacobi in the 1840s. It is not usually the quickest way to solve a particular problem. Its importance is what it revealed.
Fronts and paths
The equation has a picture attached, and the picture is the point.
Drop a stone in a pond. Ripples spread out in rings. The rings are wavefronts.
Now draw arrows pointing outward, at right angles to the rings, showing the direction the disturbance travels. Those are rays.
Fronts and rays are two descriptions of one thing. Give the fronts and the rays follow, since rays cross fronts at right angles. Give the rays and the fronts follow.
Hamilton's equation is the equation for the fronts. The trajectories of mechanics are the rays.
So a moving particle can be described either as a thing following a path, or as one ray of a family of surfaces sweeping through space.
The same equation twice
Here is what makes this more than a picture.
Optics already had an equation for wavefronts of light travelling through material that bends it.
Hamilton's equation for mechanical motion is the same equation. Not similar. The same, with one substitution: whatever plays the role of the bending power of the material corresponds to the energy available to the particle.
A particle rolling through a landscape of hills and valleys behaves exactly like light through glass of varying density.
Hamilton knew this because he did the optics first. His mechanics is a transcription of it.
The gap
Now the observation that matters, and it went unexploited for ninety-two years.
Fronts and rays are not the whole of optics. They are the approximation that works when the wavelength is very small compared with everything else. Rays are what waves look like when the waves are fine enough that you cannot see them waving.
When the wavelength is not small, rays fail. Light bends round corners, spreads through small openings, and produces interference patterns. None of that is in the ray picture.
So: if mechanics is the same as ray optics, and ray optics is an approximation to wave optics, then mechanics ought to be an approximation to a wave mechanics too.
The logic is not subtle. It is available to anyone who notices the two equations are identical.
Why nobody took it up
Nothing suggested it was worth pursuing.
To have waves you need a wavelength, and nobody had seen the slightest hint of a wavelength for a moving object. There was no experiment that ray mechanics failed to explain. There was nothing to be waving.
The analogy sat complete and idle from 1834 to 1926.
Filling it in
Schrödinger took it up in 1926, and worked backwards.
Suppose the wave exists. Suppose classical mechanics is what it looks like when the wavelength is negligible. What equation must the wave obey?
Running that requirement through the mathematics produces one answer, and it is the equation at the foundation of quantum mechanics.
It was not guessed. It was constructed, by taking Hamilton's analogy seriously and insisting that the missing corner of the table be filled in.
The wavelength turns out to be tiny for anything you can see, and comparable to the size of an atom for an electron. That is why the ray picture works so well for cricket balls and fails completely for electrons.
A place where both pictures fail
One more thing, and it connects back to earlier pages.
Where rays come back together — at a focus — the front folds over on itself and the description breaks down. Several rays arrive at one place, and the simple picture of one front passing through cannot hold.
Those are the conjugate points of Where the minimum fails, seen from the wave side.
The ray picture predicts infinite brightness at a focus, which is wrong. The wave picture gives a finite, bright, structured spot. The ray description is worst exactly where the interesting thing happens.
That pattern is general. The approximation fails precisely at the places the full theory finds most significant.
Where to go next
Hamilton's own route to all of this started from a different idea: treat the total cost as a function of where the journey starts and ends. It is at Action from the endpoints.