Compare hydrogen 1s, 2pz and 3dz² shapes. A signed-amplitude lattice reveals opposite wavefunction signs and angular nodes, while an animated slicing plane exposes cross-sections.
• A nucleus with coordinate axes, signed-amplitude regions, angular nodal surfaces, a moving section plane and a section amplitude profile. • Controls for hydrogen orbital, section offset (a₀), section scan angular rate (rad/s) and relative density threshold (fraction of peak |ψ|²). • Four tabs: Quantum bench, Probabilities & measurements, Experiments (with built-in model checks), and Learn & assess.
Unnormalized shapes in units of a₀: ψ1s ∝ exp(−r); ψ2pz ∝ z exp(−r/2); ψ3dz² ∝ (2z² − x² − y²) exp(−r/3). The number of angular nodes is l, and radial nodes are n − l − 1 (zero for these examples). Colors show sign; the threshold applies to |ψ|² relative to its maximum.
The angular and radial functional shapes are exact up to normalization constants, but the display uses a finite sampled lattice and relative-density threshold, not a specified enclosed-probability isosurface. The scene magnification changes between states. There is no spin-orbit splitting, multi-electron screening or electron trajectory.
No. The sign belongs to the amplitude; the squared magnitude is nonnegative.
No. They sample a stationary field on a lattice.
For 2pz, the z = 0 plane is a node and the entire section amplitude there is zero. For 3dz², opposite signs separate across two cones, giving the dz² lobes and the equatorial region.
The shapes are exact up to normalization, but the display is a finite sampled lattice with a relative-density threshold rather than an enclosed-probability isosurface, and it omits spin-orbit splitting and multi-electron screening.