particleinabox

Particle in a box, or the infinite potential well, is a paradigmatic model in quantum mechanics that shows how confinement leads to discrete energy levels and nontrivial wavefunctions. It describes a single nonrelativistic particle of mass m confined to a one-dimensional region 0 < x < L, with a potential V(x) = 0 inside and V(x) = ∞ outside. The walls enforce boundary conditions ψ(0) = ψ(L) = 0, and the problem is exactly solvable.

Solving the time-independent Schrödinger equation gives eigenfunctions ψ_n(x) = sqrt(2/L) sin(nπx/L) and corresponding energies E_n = n^2 π^2 ħ^2 / (2 m L^2), with n = 1, 2, 3, ... The ground state energy is nonzero due to confinement. The probability densities are nonuniform, and the wavefunctions have n−1 nodes inside the box.

In three dimensions, for a rectangular box with sides Lx, Ly, Lz, the separable solutions yield E_nml = (ħ^2/2m) [ (n_x π / Lx)^2 + (n_y π / Ly)^2 + (n_z π / Lz)^2 ] and ψ_nml(x,y,z) = (√(8)/(√(Lx Ly Lz))) sin(n_x π x/Lx) sin(n_y π y/Ly) sin(n_z π z/Lz). For a cube of side L, this reduces to E_nml = (ħ^2 π^2 / 2m L^2) (n_x^2 + n_y^2 + n_z^2). Degeneracies occur when different triplets yield the same sum.

The model is used to illustrate quantum confinement and energy quantization, with applications to teaching, quantum dots, and nanoscale systems. Realistic systems involve finite wells and interactions, and alternative geometries (cylindrical, spherical) produce different spectra.