The dumbbell and the ring

Open a chemistry textbook and the p orbitals are dumbbells: p_x along one axis, p_y along another, p_z along the third. Open a physics textbook on the hydrogen atom and the same energy level is populated by rings — states of definite angular momentum, labeled m. Both pictures are right, and the widget below is the machine that converts one into the other. Two knobs: θ mixes p_x into p_y with real coefficients, and φ gives the mix a relative phase. Everything this article claims, you can check with your own eyes before reading on.

mix θ — p_x ↔ p_y0°

phase φ of the p_y part0°

Bohmian tracersphase colors

φ = 0: a real wavefunction — the dumbbell rotates with θ, and the current vanishes identically.

The mix is a rotation

Start with φ = 0 and slide θ from 0° to 90°. The dumbbell turns, rigidly, from the x-axis to the y-axis, never changing shape. That is not an animation trick; it falls straight out of what a p orbital is. In units with ℏ = m = ω = 1, the first excited shell of the 2D harmonic oscillator is

$p_x = x\,g(r), \qquad p_y = y\,g(r), \qquad g(r) = \sqrt{\tfrac{2}{\pi}}\,e^{-r^2/2},$

— each one a plane function (x or y, a tilted plane through the origin) multiplied by a clamp (the round Gaussian g) that kills it far from the nucleus. The clamp is circularly symmetric, so it cannot participate in the shape. The real superposition is

$\cos\theta\,p_x + \sin\theta\,p_y = \big(x\cos\theta + y\sin\theta\big)\,g(r) = u\,g(r),$

where u is just the coordinate measured along the axis tilted by θ. Rotate your head by θ and the state is x′g(r) again — the same dumbbell, because the space of p states closes under rotation. The strip under the widget shows the pieces: the round clamp, the plane, and the cross-section of their product along the mix axis. Every picture of a “tilted p orbital” you have ever seen is this one line of algebra.

The phase closes the ring

Now the second knob. Hold θ = 45° — an equal mix — and slide φ from 0° to 90°. The dumbbell fattens, its waist fills in, and at φ = 90° the density is a ring: circularly symmetric, with a dead center. The algebra is the same line with a phase attached:

$\frac{p_x + i\,p_y}{\sqrt{2}} = \frac{(x + iy)}{\sqrt{2}}\,g(r) = \frac{r\,e^{i\varphi}}{\sqrt{2}}\,g(r),$

so $|\Psi|^2 = \tfrac12 r^2 g(r)^2$, which depends on r alone. The density forgot the angle entirely — but the wavefunction did not: its phase is $\varphi$, winding once around the ring. Turn on the widget’s phase colors and you see it directly: hue is arg Ψ, and around the ring the hue cycles through the full rainbow exactly once. That winding is the angular momentum. The dumbbell, in this language, is two opposite windings standing on top of each other — counter-rotating rings superposed with real weights, their currents canceling, leaving lobes. p_x and p_y are the standing-wave basis; the rings are the traveling-wave basis. Same shell, same energies, different choice of what to call simple.

Phase gradient is velocity

And that winding is not bookkeeping. Watch the amber tracers with φ = 0: they stand still, because a real-valued wavefunction carries identically zero probability current — the same verdict that convicted the 2011 animation in the superposition autopsy. Now raise φ: the tracers begin to circulate, fastest at φ = 90°, inner ring outrunning the outer. The Schrödinger equation reads a spatial phase variation as a flow of probability:

$\mathbf{j} = \operatorname{Im}\big(\Psi^*\nabla\Psi\big) = \rho\,\nabla S,$

so the Bohmian velocity $\mathbf{j}/\rho = \nabla S$ is literally the gradient of the phase — a positional phase variation is a probability current. On the ring, S = φ and $\nabla S = (-y, x)/r^2$: solid circulation with a 1/r shear, the same formula the atomic orbitals demo integrates in 3D for its m > 0 states. The density is perfectly stationary — the ring does not precess — yet every tracer is moving. That is the whole content of “stationary state with angular momentum”: the cloud stands still and the probability flows through it, everywhere at once, forever. The widget’s tracers are Born-sampled from ρ and advected by ∇S, so they are the density carried along — the continuity equation guarantees the ensemble stays Born-distributed as it circulates.

One more honesty note, the same one the autopsy made: none of this says an electron is a little planet. The tracers are what a weak momentum measurement returns, not hidden gears — but the current they reveal is as real as the interference that built it, and it is the reason the m states, not the dumbbells, are the ones that couple to a magnetic field. That is a story for the Zeeman article.