A short, wave-based derivation of energy quantization for an infinite square well.
Consider a particle confined to a one-dimensional region of length L with infinitely high walls at x = 0 and x = L. The particle cannot exist outside the box, so its wave must be zero at the walls: ψ(0) = 0 and ψ(L) = 0.
Using de Broglie relation, a particle with momentum p has wavelength λ = h/p. Inside the box the wave behaves like a standing wave, because reflections from the walls superpose the traveling waves. The boundary conditions force an integer number of half-wavelengths to fit exactly into the length L:
L = nλ/2, n = 1, 2, 3, …
Therefore the allowed wavelengths are λn = 2L/n. Substituting into de Broglie relation gives quantized momentum:
pn = h/λn = nh/(2L).
For a nonrelativistic particle, the kinetic energy is E = p2/(2m). Hence the allowed energies are
En = pn2/(2m) = (n2h2)/(8mL2), n = 1, 2, 3, …
This is the key result: confinement plus the standing-wave condition produces discrete wavelengths, which implies discrete momenta and therefore discrete energy levels.