An electron confined to a 1D box of length L has only discrete allowed energies, from solving the Schrödinger equation with infinite walls:
E_n = n²h² / (8 m_e L²)
= n² · 0.3759 eV·nm² / L²(nm)
ψ_n(x) = √(2/L) · sin(nπx/L)
Each level n is a single spatial state, so the Pauli exclusion principle allows at most two electrons there — one spin-up (↑), one spin-down (↓), the only way two electrons can share every other quantum number. Electrons fill from n = 1 upward (Aufbau): the ladder on the left shows every level up to the LUMO, spaced by n itself while the printed value is the real E_n; filled rungs carry their spin-pair arrows, the topmost filled rung is the HOMO (yellow), the first empty one above it is the LUMO (dim).
- Shrink the box — every E_n grows as 1/L², so squeezing the same electrons apart in energy is the direct 1D analogue of the degeneracy pressure that holds up a white dwarf.
- Add electrons — each new pair must occupy a higher, unoccupied level; there is no way to add a third electron to a full n without violating Pauli.
- Right pane — the probability density |ψ(x)|² for the HOMO level across the box, showing where that electron is most likely to be found.