Wavefunctions in Three Dimensions

Open any chemistry textbook and the 2p orbital is a pair of glossy lobes with a crisp skin. That skin is not in the physics. It is the surface inside which 90% (or 95%, or whatever the illustrator chose) of the probability lives, and the wavefunction continues smoothly through it and out the other side. Drawing the surface is a decision to hide the tail and to hide the fact that the interior is not uniform either.

This page ray-marches |ψ|² itself: every voxel glows and absorbs in proportion to how likely the particle is to be there, and what reaches your eye along each line of sight is the accumulated probability. Nothing is thresholded unless you ask for it — raise the transparency floor and watch the isosurface picture reappear as a special case, the one that happens when you decide the faint parts do not count.

Step a quantum number. In the box and the oscillator each axis has its own integer, and each integer adds one nodal plane perpendicular to that axis: the nodes readout counts them along a line through the cloud. Turn on the sign colouring and the planes are where rose meets sky.

Click through the states that share an energy. The energy readout does not change. The shape does, sometimes dramatically: in a cube, (1,1,2), (1,2,1) and (2,1,1) are the same object pointing three ways, and (3,3,3) shares its energy with (5,1,1) for no symmetry reason at all — 27 = 27 is an accident of arithmetic. Then switch the cube to a brick, or the trap to the squashed one, and watch the partner list shrink as the symmetry that protected the degeneracy is taken away.

Slice the cloud. The 3s orbital and the (0,0,2) oscillator state hide their structure inside an outer shell. Turn on the clip plane and drag it through the cloud — the spherical nodal shells appear as dark gaps the density never crosses.

The isotropic oscillator at level N = nₓ + nᵧ + n_z holds ½(N+1)(N+2) states: 1, 3, 6, 10. The hydrogen shell at n holds n² of them (before spin). The rigid rotor at l holds 2l + 1. In each case the count is a symmetry counting itself — rotations of the trap, the extra hidden symmetry of the Coulomb problem, rotations of the sphere — and in each case the degenerate states are as different in shape as states get.

The rotor makes the sharpest point. In the definite m basis the density has no dependence on the angle around z at all: Y₂¹ is a smooth belt, and only the phase (the hue) knows it is spinning. Switch to the real lobes basis — the same energy, the same l, a different choice of which combinations to draw — and the belt becomes four lobes with nodal planes between them. Both pictures are right. Neither is the shape of the state, because a degenerate level does not have one.

You have been looking at the box state (1, 1, 2), with energy 3.000 π² ħ²/mL² and 2 other states at exactly the same energy. The grid integral of |ψ|² is 1.000: the cloud you rotated holds 100% of the particle, and the rest is in the tail outside the box, faint but not zero — which is exactly the part an isosurface would have thrown away.

If the browser could not create a WebGL2 context, the same data was shown as a single slice on a 2-D canvas, with the same colours and the same clip position. The nodes are just as countable in a slice; only the sense of depth is lost.