The ladder of scales
Every scale, live
Seven rungs, seven real solvers. Climb from a wavefunction to a pendulum on your desk.
~10⁻¹⁰ m · a wavepacket smaller than an atom
Quantum wavepacket
A single quantum particle drifts into a barrier — and part of it goes straight through.
ModeA single quantum particle drifts into a barrier — and part of it goes straight through. the maths
Solved by split-step Fourier: half a step of the potential, a Fourier-space half step of the kinetic part, half a step of the potential again. The barrier sits just above the packet's mean energy, so you watch tunnelling and reflection at once.
~10⁻¹⁰ m · the atom itself
Hydrogen orbital
One electron around one proton. Every dot is where a measurement might find it — watch the orbital emerge from the noise.
OrbitalOne electron around one proton. Every dot is where a measurement might find it — watch the orbital emerge from the noise. the maths
Exact analytic orbitals sampled as |ψ|² point clouds: the radius comes from an inverse-CDF of r²R², the direction from rejection sampling on |Y|². Gold and violet mark the two signs of the real ψ — the nodal plane between lobes is where the wavefunction crosses zero.
~10⁻⁹ m · molecules touching
Lennard-Jones melting
Atoms held together by the Lennard-Jones force freeze into a crystal, melt into a droplet, then boil away — just turn the heat up.
Atoms held together by the Lennard-Jones force freeze into a crystal, melt into a droplet, then boil away — just turn the heat up. the maths
Velocity-Verlet dynamics in reduced units, the potential cut and shifted at 2.5σ, and a Berendsen thermostat. The order readout is ⟨|ψ₆|⟩, the local six-fold bond-orientation order parameter — near 1 for a crystal, collapsing when it melts.
~10⁻⁶ m · a breath of air
Kinetic gas
Hundreds of colliding discs settle into the Maxwell–Boltzmann speed distribution on their own.
Hundreds of colliding discs settle into the Maxwell–Boltzmann speed distribution on their own. the maths
Every collision is elastic; nothing is drawn from the curve. Starting all discs at the same speed, the histogram still relaxes to the thin cream line — equilibrium is emergent. The piston wall does real work, so compressing the gas genuinely heats it.
statistical · many particles become a probability field
Particles → Probability
Thousands of Brownian particles jiggling in a double well. Fade the dots out and what remains is the probability density itself.
Thousands of Brownian particles jiggling in a double well. Fade the dots out and what remains is the probability density itself. the maths
Euler–Maruyama overdamped Brownian dynamics — nothing about the equilibrium shape is drawn in by hand. The cloud is a Gaussian-kernel density estimate of where the particles actually are, which relaxes to the Boltzmann distribution on its own.
~10⁻² m · a stirred fluid
Shear-flow fluid
Two layers of water sliding past each other fold into Kelvin–Helmholtz whirlpools. Drag to stir.
Two layers of water sliding past each other fold into Kelvin–Helmholtz whirlpools. Drag to stir. the maths
Stam's stable-fluids scheme: semi-Lagrangian advection, a Jacobi pressure projection to keep the flow divergence-free, and a little vorticity confinement so the swirls survive numerical damping. Seeded with a tanh shear layer along the diameter.
~1 m · everyday objects
Pendulum phase space
A pendulum swings on the left while its (θ, p) orbit draws itself on the right — orbit, separatrix and all.
A pendulum swings on the left while its (θ, p) orbit draws itself on the right — orbit, separatrix and all. the maths
Leapfrog integration — symplectic without damping, so the orbit neither spirals nor blows up. The ember curve on the phase plane is the separatrix: inside it the pendulum swings, outside it loops over the top. Drag the bob, or tap the plane to drop the state anywhere.