Trap an electron inside a circular wall and its wavefunction has nowhere to go but into standing waves — exact rings of probability set by the zeros of a Bessel function. This is the shape behind the famous scanning-tunnelling image of 48 iron atoms on copper: the electron, bound, rendered as a relief of its own .
TIER A · DIRECT the surface is the exact eigenstate of the circular infinite well — — nothing sketched.A particle confined to a disk of radius must vanish at the wall, exactly as a drumhead must be still at its rim. That single boundary condition quantizes it: the allowed states are , where is the -th zero of the Bessel function . The integer n counts the concentric rings; m counts the angular lobes. The height below is the electron's probability density — the ripples are where it is likely to be found.
A companion to Quantum Wells & Tunnelling: the same confinement-quantizes-a-standing-wave idea, taken from one dimension into two. An electron on a disk has eigenstates fixed by the zeros of Bessel functions, and its probability density is a relief of concentric rings — the shape made iconic by the 1993 IBM quantum-corral image.
The rendered surface is for the 2-D circular infinite well, . The Bessel zeros , the energies , and the node counts are computed and cross-checked against SciPy; is evaluated in-page by its integral representation.
A perfectly hard circular wall and a single non-interacting electron. The real corral is a leaky ring of adatoms and the STM sees the surface local density of states — the same standing-wave physics with a soft wall. The confining “atoms” drawn around the rim are schematic markers of the wall, not computed adatom potentials.