physics

The dumbbell and the ring

A p orbital is a plane times a clamp, and mixing p_x with p_y just rotates the plane. Give the mix a phase and the dumbbell closes into a ring with a wound phase — and the Schrödinger equation reads that winding as a probability current, made visible by a cloud of weakly sampled tracers.

A superposition is not a morph

For fifteen years the harmonic-oscillator demo on this site animated a superposition as cos t·ψ_A + sin t·ψ_B and called it time evolution. It was not — and the conviction doesn't even need the Rabi oscillation it happens to resemble: a real wavefunction carries identically zero probability current, so the morphing density was teleporting. The original animation, preserved as the bug it was, with the Born flow that convicts it.

From exponential growth to the logistic curve

A single population grows without bound, predators can only cage it into a boom-and-bust cycle, and the honest fix is a population that limits itself: the S-curve.

Van der Pol and the birth of a limit cycle

An oscillator with friction that changes its mind. Small swings get pumped up, big ones get bled down, and everything meets on a single closed loop the whole plane falls onto.

The pendulum, the cat's eye, and a little friction

The pendulum is the spring with the small-angle lie removed. Putting the full sine back in bends its phase space into the cat's-eye pattern, and friction drains it.

The spring equation and the shape of simple motion

Where the spring equation comes from, and why its phase space is a family of nested ellipses that damping turns into a spiral.

Drawing differential equations as wind

An old widget of mine, rebuilt in plain javascript, that draws the phase space of a differential equation the way those wind maps draw the wind.

Current project

All things must pass

I never knew it looked like that

Electron sausages?

Fuzzy and indistinct

I think I've decided

This is one of the coolest things I've seen in quite a while