Lab 10 — Quantum Mechanics

The Square Well (Particle in a Box)

This is the first problem every quantum course solves: a particle trapped in a box of width L. In quantum mechanics a particle is also a wave, so only standing waves that fit exactly between the walls can exist — which limits the energy to discrete values (quantization); the exact form is the formula just below. Just as a guitar string sounds only its harmonics, this is why an electron’s energy comes in discrete rungs rather than a continuum.

\psi_n(x) = \sqrt{\dfrac{2}{L}}\,\sin\!\left(\dfrac{n\pi x}{L}\right) \qquad E_n = \dfrac{n^2 \pi^2 \hbar^2}{2mL^2} \qquad n = 1,\, 2,\, 3,\, \dots

Level nn = 1

Box width LL = 1.00

EnergyE₁ = 1.00

Three lessons live here. (1) Energy is quantized — no value between the rungs is allowed. (2) Even the ground state E₁ is nonzero (zero-point energy): by the uncertainty principle a confined particle can never be fully at rest. (3) Shrinking the box raises every energy as 1/L² — this is why the color of a semiconductor quantum dot depends on its size. Notice too that higher n means more nodes — points where the wave stays pinned at zero — and a faster shimmer. Switch to |ψ|² and the shape freezes: a stationary state’s probability distribution never changes.