Editing Uncertainty principle (section)

===Wave mechanics interpretation===
{{Main article|Wave packet|Schrödinger equation}}
{{multiple image
| align = right
| direction = vertical
| footer = Propagation of [[matter wave|de Broglie waves]] in 1d—real part of the [[complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the colour [[opacity (optics)|opacity]]) of finding the particle at a given point ''x'' is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the [[curvature]] reverses sign, so the amplitude begins to decrease again, and vice versa—the result is an alternating amplitude: a wave.
| image1 = Propagation of a de broglie plane wave.svg
| caption1 = [[Plane wave]]
| width1 = 250
| image2 = Propagation of a de broglie wavepacket.svg
| caption2 = [[Wave packet]]
| width2 = 250
}}
According to the [[Matter wave|de Broglie hypothesis]], every object in the universe is associated with a [[wave]]. Thus every object, from an elementary particle to atoms, molecules and on up to planets and beyond are subject to the uncertainty principle.

The time-independent wave function of a single-moded plane wave of wavenumber ''k''<sub>0</sub> or momentum ''p''<sub>0</sub> is<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 60 | bibcode = 2013qtm..book.....H }}</ref>
<math display="block">\psi(x) \propto e^{ik_0 x} = e^{ip_0 x/\hbar} ~.</math>

The [[Born rule]] states that this should be interpreted as a [[probability density function|probability density amplitude function]] in the sense that the probability of finding the particle between ''a'' and ''b'' is
<math display="block"> \operatorname P [a \leq X \leq b] = \int_a^b |\psi(x)|^2 \, \mathrm{d}x ~.</math>

In the case of the single-mode plane wave, <math>|\psi(x)|^2</math> is ''1'' if <math>X=x</math> and ''0'' otherwise. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet.

On the other hand, consider a wave function that is a [[superposition principle|sum of many waves]], which we may write as
<math display="block">\psi(x) \propto \sum_n A_n e^{i p_n x/\hbar}~, </math>
where ''A''<sub>''n''</sub> represents the relative contribution of the mode ''p''<sub>''n''</sub> to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized. We may take this a step further to the [[continuum limit]], where the wave function is an [[integral]] over all possible modes
<math display="block">\psi(x) = \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^\infty \varphi(p) \cdot e^{i p x/\hbar} \, dp ~, </math>
with <math>\varphi(p)</math> representing the amplitude of these modes and is called the wave function in [[momentum space]]. In mathematical terms, we say that <math>\varphi(p)</math> is the ''[[Fourier transform]]'' of <math>\psi(x)</math> and that ''x'' and ''p'' are [[conjugate variables]]. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta.<ref name="L&L">{{cite book |first1=Lev Davidovich |last1=Landau|authorlink1=Lev Landau|first2=Evgeny Mikhailovich|last2=Lifshitz|authorlink2= Evgeny Lifshitz|year=1977 |title=Quantum Mechanics: Non-Relativistic Theory |edition=3rd |volume=3 |publisher=[[Pergamon Press]] |isbn=978-0-08-020940-1|url=https://archive.org/details/QuantumMechanics_104}}</ref>

One way to quantify the precision of the position and momentum is the [[standard deviation]]&nbsp;''σ''. Since <math>|\psi(x)|^2</math> is a probability density function for position, we calculate its standard deviation.

The precision of the position is improved, i.e. reduced ''σ''<sub>''x''</sub>, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased ''σ''<sub>''p''</sub>. Another way of stating this is that ''σ''<sub>''x''</sub> and ''σ''<sub>''p''</sub> have an [[inverse relationship]] or are at least bounded from below. This is the uncertainty principle, the exact limit of which is the Kennard bound.