Fluid and Wave-Packet Pictures

Madelung's rewriting

Write the wavefunction in polar form — an amplitude and a phase — substitute into the Schrödinger equation, and separate real and imaginary parts. What emerges is a continuity equation and something that looks exactly like Euler's equation for an irrotational fluid, with one extra term: the quantum potential, built from the curvature of the amplitude. This is an exact rewriting, not an approximation. Nothing has been added and nothing lost.

The pictures that follow are genuinely useful. Probability current becomes a flow field. Nodal lines of the wavefunction become vortices with quantised circulation, because the phase must return to itself modulo 2π around any closed loop. Interference becomes a pattern of streamlines. Students see structure they would not have seen in a table of amplitudes.

Why it is not a fluid

The quantum potential is non-local in the sense that matters: it depends on the shape of the amplitude everywhere, not on local density and pressure. There is no equation of state. The 'fluid' of a two-particle system lives in six dimensions, not three, and no rearrangement recovers a three-dimensional flow. And the streamlines are not trajectories of anything that has been observed to travel along them; they are integral curves of a current density, which is a different kind of object.

The reason the chapter exists is that the visualisation is beautiful, and beauty is what tempts the substitution. A rendered flow field looks like evidence. It is a coordinate change.

Rydberg packets and the stadium

Two concrete systems keep this from being abstract. A Rydberg wave packet — a superposition of highly excited atomic states — is localised enough to be discussed as an orbiting blob, and it does orbit, and it also disperses and revives on a schedule the classical picture cannot supply. The classical orbit is a proxy with a stated lifetime.

The Bunimovich stadium is the other. Classically its billiard dynamics are chaotic; quantum mechanically there are no trajectories to be chaotic, only a spectrum whose statistics match those of random matrices, and eigenfunctions that scar along unstable periodic orbits. The classical orbit shows up as a feature of a wave, which is precisely the relationship this volume keeps trying to describe: the borrowed picture leaves a fingerprint on the real object without being the real object.

Solved, but never filmed

A recent post put the distinction in one sentence: the flow around a wing was never filmed; it was solved. Computational fluid dynamics advances the Navier-Stokes equations through time and returns pressure, lift, and vortices rolling from the edges. The resulting animation looks like a recording of air moving over a wing, but it is not. It is a rendering of a numerical solution to a set of partial differential equations, given boundary conditions, a turbulence model, and a mesh.

That is not a dismissal. CFD is one of the most successful applied-mathematics programmes of the last century. Aircraft are designed with it, weather is forecast with it, blood flow in arteries is planned with it. The point is epistemological, not technical. A solved flow field is a proposition about the world: if the equations are right, if the boundary conditions match the experiment, if the turbulence model captures the scales that matter, then the field at the next time step will look like this. A film is a different proposition: light scattered by actual fluid elements arrived at a sensor in this pattern. The two can agree and still be different kinds of evidence.

The temptation is to treat the solved image as more real than the filmed one because it is cleaner. The computation removes camera noise, lighting irregularities, seeding density, and the limits of human reaction time. It also removes the check that the model had to survive. Smoke-tunnel photographs and particle-image velocimetry of wings do exist; they are messier and more partial, but they are what the solution must be compared against. The phrase 'never filmed' is therefore best read as a rhetorical emphasis on the power of the method, not as a literal claim that no optical measurement of wing flow has ever been made.

This is the same discipline the chapter asks for everywhere else. A Madelung streamline is a solved object, not a trajectory. A Rydberg wave packet is a solved object, not a planet. The CFD wing is simply the example that makes the point visible to someone who has never opened a quantum textbook. The visualisation is beautiful, and beauty is still the warning sign: the moment a solved picture is mistaken for a direct observation, the proxy has become the evidence.