Particle in a Box: Energy Levels and Wave Functions
The Schrodinger equation, solved for the simplest trap there is. Set the box width and the quantum number and watch the wave function gain exactly one more half-wave — and one more interior node — per step up, while the energy ladder beside it climbs as n squared.
Energy of this level E = n²h² / 8mL²
3.384 eV
5.421 × 10-19 J
Box width L1.00 nm
Quantum number n3
Gap to the level above E(n+1) − E(n)2.368 eV
Gap down to n − 1 (2n−1)h² / 8mL²1.880 eV
Photon emitted falling to n − 1659.5 nmvisible light
De Broglie wavelength 2L / n0.667 nm
Interior nodes n − 12
Half-waves in the box3
Particle: electron · mass 9.109 × 10-31 kg h = 6.62607015 × 10-34 J·s · walls infinitely high
Tip: hold n fixed and halve the box width. Every level jumps by a factor of four, because L is squared and sits underneath.