Editing Uncertainty principle (section)

===Quantum harmonic oscillator stationary states===
{{Main article|Quantum harmonic oscillator|Stationary state}}
Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in terms of the [[creation and annihilation operators]]:
<math display="block">\hat x = \sqrt{\frac{\hbar}{2m\omega}}(a+a^\dagger)</math>
<math display="block">\hat p = i\sqrt{\frac{m \omega\hbar}{2}}(a^\dagger-a).</math>

Using the standard rules for creation and annihilation operators on the energy eigenstates,
<math display="block">a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle</math>
<math display="block">a|n\rangle=\sqrt{n}|n-1\rangle, </math>
the variances may be computed directly,
<math display="block">\sigma_x^2 = \frac{\hbar}{m\omega} \left( n+\frac{1}{2}\right)</math>
<math display="block">\sigma_p^2 = \hbar m\omega \left( n+\frac{1}{2}\right)\, .</math>
The product of these standard deviations is then
<math display="block">\sigma_x \sigma_p = \hbar \left(n+\frac{1}{2}\right) \ge \frac{\hbar}{2}.~</math>

In particular, the above Kennard bound<ref name="Kennard" /> is saturated for the [[ground state]] {{math|''n''{{=}}0}}, for which the probability density is just the [[normal distribution]].