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A wave packet in hydrogen

Hydrogen Exactly

Computed equatorial density section of the initial n=24 to 36 circular-state packet, localized on the left of the nucleus.
Circular packetDensity false color, cyan to gold

Drag with a mouse or one finger to rotate the 3D view. Arrow keys rotate when the image has focus. Home resets the camera. Hover a density contour to read its meaning. Click or tap to pin its explanation and highlight the layer. The selection stays pinned if the layer disappears and highlights it again when it returns. Clear, Escape, or a click on empty space releases it. Instruments has six layer buttons that can also pin a layer for keyboard use. Rotation is available in the contour and soft volume views.

Preparing the wavefunction

13 circular states, n = 24 to 36Characteristic radius 47.6 nm
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A localized hydrogen wave packet spreads around its orbit, forms smaller packets, and partly gathers itself again. The colored contours mark increasing probability density, from cyan through violet and pink to gold. This image comes from adding 13 circular states and evolving each with its own energy. Switch to the low-n mode in Instruments to see interference frequencies encoded as the familiar hydrogen colors. Those colors describe beats in the density; they do not show emitted light.

Hover a layer to read its density. Click or tap to pin the explanation and highlight that layer. The selection stays pinned if the layer disappears, then highlights it again when it returns; Clear or Escape releases it. All the layers belong to the same electron. Each surface marks three times the density of the previous one: in equally small regions, a higher density means a greater chance of detecting the electron. A layer appears only where that density is reached. The layers are not separate energy shells.

The ring comes from the highly excited circular states chosen for this packet. Their density is concentrated around an equatorial ring; the packet initially occupies an arc and later spreads around it. Hydrogen's lowest-energy 1s state has a spherical probability cloud. The shape depends on the quantum state, rather than every hydrogen atom having a doughnut shape.

What “exactly” means here, and sources

These are exact eigenstates of the fixed-nucleus, nonrelativistic Coulomb model, evaluated numerically. The finite state list, render box, f32 shader arithmetic, f16 display texture, grid interpolation, and ray samples introduce errors. Nuclear motion, relativity, and radiative corrections are omitted. The 2D section evaluates the wavefunction directly; the volume interpolates sampled positive density.

The default view draws surfaces at six fixed density levels, revealing the highest contour reached along each view ray. Directional lighting, highlights, shadows, a faint reference plane, and the cyan-to-gold palette help show the shape. These are visual aids; the plane is not part of the atom. They leave the wavefunction and its diagnostics unchanged. Soft volume and a directly evaluated 2D section are also available in Instruments. Spectral color weights the magnitudes of visible signed cross terms. A separate diagnostic marks the sign of the 2p / 3s term. Dark nodes have zero analytic density; volume interpolation can soften them.

  1. Bluhm, Kostelecký, and Porter, wave-packet evolution and revivals. Characteristic scales guide the event search; the actual energies determine the image.
  2. MIT, hydrogen atom lecture notes. Quantum numbers and the spherical 1s ground state.
  3. NIST Bohr radius and Rydberg constant. The circular n = 30 radius is about 48 nm. The low-n view uses a separate scale.
  4. NIST measured hydrogen line components. Visible air values are separate from this ideal model's vacuum wavelengths.
  5. CIE 1931 observer table (CC BY-SA 4.0). XYZ is converted to linear sRGB, negative channels clipped, and the maximum channel normalized to one. Gamut clipping and exposure mean the screen is an encoding.
  6. Model, numerical checks, room interface, and measured limits.