Uncertainty principle
Particle in a box

Consider a particle in a one-dimensional box of length L {\displaystyle L}. The eigenfunctions in position and momentum space are ψ n (x, t) = { A sin (k n x) e − i ω n t, 0 < x < L, 0, otherwise, {\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin(k_{n}x)\mathrm {e} ^{-\mathrm {i} \omega _{n}t},&0<x<L,\\0,&{\text{otherwise,}}\end{cases}}} and φ n (p, t) = π L ℏ n (1 − (− 1) n e − i k L) e − i ω n t π 2 n 2 − k 2 L 2, {\displaystyle \varphi _{n}(p,t)={\sqrt {\frac {\pi L}{\hbar }}}\,\,{\frac {n\left(1-(-1)^{n}e^{-ikL}\right)e^{-i\omega _{n}t}}{\pi ^{2}n^{2}-k^{2}L^{2}}},} where ω n = π 2 ℏ n 2 8 L 2 m {\textstyle \omega _{n}={\frac {\pi ^{2}\hbar n^{2}}{8L^{2}m}}} and we have used the de Broglie relation p = ℏ k {\displaystyle p=\hbar k}. The variances of x {\displaystyle x} and p {\displaystyle p} can be calculated explicitly: σ x 2 = L 2 12 (1 − 6 n 2 π 2) {\displaystyle \sigma _{x}^{2}={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)} σ p 2 = (ℏ n π L) 2. {\displaystyle \sigma _{p}^{2}=\left({\frac {\hbar n\pi }{L}}\right)^{2}.}

The product of the standard deviations is therefore σ x σ p = ℏ 2 n 2 π 2 3 − 2. {\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}.} For all n = 1, 2, 3, … {\displaystyle n=1,\,2,\,3,\,\ldots }, the quantity n 2 π 2 3 − 2 {\textstyle {\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}} is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when n = 1 {\displaystyle n=1}, in which case σ x σ p = ℏ 2 π 2 3 − 2 ≈ 0.568 ℏ > ℏ 2. {\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {\pi ^{2}}{3}}-2}}\approx 0.568\hbar >{\frac {\hbar }{2}}.}
keywords:frequency, detectable_format:constrained_response