Particle in a Box Calculator

The particle in a box (infinite square well) is the simplest quantum mechanics model: a particle confined between two impenetrable walls of separation L. The allowed energies are En = n^2 h^2 / (8mL^2) for n = 1, 2, 3..., where h is Planck's constant, m is the particle mass, and L is the box length. Only discrete energy values are allowed because the wavefunction must be zero at the walls.

The wavefunction must satisfy boundary conditions: psi = 0 at both walls. This requires the box to contain an integer number of half-wavelengths: L = n * lambda/2, so lambda = 2L/n. The de Broglie relation p = h/lambda gives p = nh/(2L), and kinetic energy E = p^2/(2m) = n^2 h^2/(8mL^2). Only specific values of n (positive integers) satisfy the boundary conditions.

The ground state (n=1) has energy E1 = h^2/(8mL^2), which is nonzero. This is the zero-point energy: the minimum energy a confined quantum particle must have, even at absolute zero. It arises from the Heisenberg uncertainty principle: confining the particle gives it a position uncertainty, requiring nonzero momentum uncertainty, and therefore nonzero kinetic energy.

The model approximates electrons confined in: (1) conjugated molecules (pi electron systems), where the box length relates to the molecular length and gives the UV/visible absorption wavelength; (2) quantum dots, where the box is the semiconductor nanoparticle and the energy levels are tunable by size; (3) nanostructures and quantum wells in semiconductor devices.

Energy levels scale as 1/L^2. Doubling the box length decreases all energy levels by a factor of 4. This explains why quantum dots (3 to 10 nm) have size-tunable optical properties: larger dots absorb and emit lower energy (red-shifted) light. The energy spacing between levels also decreases as 1/L^2, so large boxes approach the classical continuum.