Editing Uncertainty principle (section)

===Particle in a box===
{{Main article|Particle in a box}}
Consider a particle in a one-dimensional box of length <math>L</math>. The [[Particle in a box#Wavefunctions|eigenfunctions in position and momentum space]] are
<math display="block">\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}</math>
and
<math display="block">\varphi_n(p,t)=\sqrt{\frac{\pi L}{\hbar}}\,\,\frac{n\left(1-(-1)^ne^{-ikL} \right) e^{-i \omega_n t}}{\pi ^2 n^2-k^2 L^2},</math>
where <math display="inline">\omega_n=\frac{\pi^2 \hbar n^2}{8 L^2 m}</math> and we have used the [[de Broglie relation]] <math>p=\hbar k</math>. The variances of <math>x</math> and <math>p</math> can be calculated explicitly:
<math display="block">\sigma_x^2=\frac{L^2}{12}\left(1-\frac{6}{n^2\pi^2}\right)</math>
<math display="block">\sigma_p^2=\left(\frac{\hbar n\pi}{L}\right)^2. </math>

The product of the standard deviations is therefore
<math display="block">\sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}.</math>
For all <math>n=1, \, 2, \, 3,\, \ldots</math>, the quantity <math display="inline">\sqrt{\frac{n^2\pi^2}{3}-2}</math> is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when <math>n = 1</math>, in which case
<math display="block">\sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{\pi^2}{3}-2} \approx 0.568 \hbar > \frac{\hbar}{2}.</math>