Boxanne

Meet Boxanne, a particle shut inside a box with perfectly rigid walls. She can only fit whole halves.

In quantum mechanics a particle is described by a wave. Rigid walls force that wave to be zero at both ends, just like a guitar string pinned at its bridge and nut. The only waves that fit are those with a whole number n of half-wavelengths across the box width L. A wavelength fixes a momentum through de Broglie's relation, and the momentum fixes the energy, so the allowed energies come out as E_n = n²h²/(8mL²).

Three things follow. The energy comes in steps, not a smooth range. The steps grow as n², so level 2 has four times the energy of level 1 and level 3 has nine times. And the lowest energy is not zero: a confined particle can never sit perfectly still.

The model is a first sketch of real systems such as electrons in conjugated dye molecules and in quantum dots, where shrinking the box raises the energies and shifts the colour of light absorbed or emitted.

Real walls are never infinitely high, so real particles leak a little into them. The infinite box is the idealised version, and Boxanne likes it that way.