Partícula na Caixa — Energia

Particle in a box: E_n = n²·h²/(8mL²)

Trap a particle of mass m inside a 1D infinite well of width L and its energy can only take certain values, E_n = n²·h²/(8mL²), where n runs through 1, 2, 3 and so on. There is no n = 0. What surprises most people is that the ground state E_1 sits above zero. That floor is the zero-point energy, and it follows straight from Heisenberg's uncertainty principle, since pinning down Δx forces Δp above zero. Because the levels go as n², the gaps between them widen the higher you climb. Take an electron (m = 9.11·10⁻³¹ kg) in a box of L = 1 nm: its E_1 lands near 6.0·10⁻²⁰ J, roughly 0.376 eV. Shrink L tenfold and E jumps by a factor of 100. That sensitivity to size is exactly why quantization only shows up at the nanoscale. Make the box big and the levels pack together so tightly they blur into a continuum.

Applications

You meet this model in quantum wells (QW) inside semiconductor lasers and high-brightness LEDs, in quantum dots (QLED TVs, fluorescent biological markers), and in conjugated polymers, where the free-electron picture describes the π-electron system. It also shows up in molecular orbital theory. And it opens the first chapter of nearly every quantum mechanics textbook, mostly because the math actually has a clean, exact solution.

FAQ

Why can't n = 0? Plug in n = 0 and you get ψ ≡ 0 everywhere, which describes no particle at all. Between the boundary conditions ψ(0) = ψ(L) = 0 and the need to normalize, n has to be at least 1.

What does zero-point energy mean physically? Cool the system down to T = 0 K and the particle still won't sit still. Being confined within L forces a nonzero minimum kinetic energy, so E_1 stays above 0.

Does this work in 3D? It does. The energy turns into a sum, E = (h²/8m)(n_x²/L_x² + n_y²/L_y² + n_z²/L_z²), so now you carry three quantum numbers. When the box is a cube, different combinations can land on the same energy, giving you degeneracies.