Volume 27 · Part Two · Chapter 4 of 19
Fluid and Wave-Packet Pictures
Madelung streamlines, Rydberg wave packets, and the Bunimovich stadium: continuum pictures laid over objects that are not continua.
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.
Equations borrowed
- ψ = √ρ · e^{iS/ħ} (Madelung transformation)
- Continuity equation ∂ρ/∂t + ∇·(ρv) = 0, with v = ∇S/m
- Quantum potential Q = −(ħ²/2m)·(∇²√ρ)/√ρ
- Random-matrix spectral statistics for classically chaotic billiards
Validity band
Exact as algebra for single-particle systems. Fails as a fluid picture wherever configuration space exceeds three dimensions, and wherever the quantum potential dominates.
Falsifier
Any experiment tracking a particle along a Madelung streamline and finding the streamline where standard quantum mechanics does not put current would break the rewriting rather than the reading.
Where this chapter is weakest
The chapter is strong on the mathematics and weaker on the sociology of the mistake: it asserts that beautiful visualisations drive substitution without documenting cases. It should carry examples from the literature rather than a general warning.
The volume-wide audit of these weak points is collected in Where This Volume Is Weak.