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{{Short description|Foundational principle in quantum physics}}
{{Other uses}}
{{Use American English|date=January 2019}}
{{Quantum mechanics}}

[[File:Werner Heisenberg - Canonical commutation rule for position and momentum variables of a particle - Uncertainty principle, 1927.jpg|thumb|Canonical commutation rule for position ''q'' and momentum ''p'' variables of a particle, 1927. ''pq'' − ''qp'' = ''h''/(2''πi''). Uncertainty principle of Heisenberg, 1927.]]

The '''uncertainty principle''', also known as '''Heisenberg's indeterminacy principle''', is a fundamental concept in [[quantum mechanics]]. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and [[momentum]], can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.

More formally, the uncertainty principle is any of a variety of [[Inequality (mathematics)|mathematical inequalities]] asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such as [[Position (vector)|position]], ''x'', and momentum, ''p''.<ref name=Sen2014>{{Cite journal | last1 = Sen | first1 = D. | title = The Uncertainty relations in quantum mechanics | url = http://www.currentscience.ac.in/Volumes/107/02/0203.pdf | journal = Current Science | volume = 107 | issue = 2 | year = 2014 | pages = 203–218 | access-date = 2016-02-14 | archive-date = 2019-09-24 | archive-url = https://web.archive.org/web/20190924115453/https://www.currentscience.ac.in/Volumes/107/02/0203.pdf | url-status = live }}</ref> Such paired-variables are known as [[Complementarity (physics)|complementary variables]] or [[Canonical coordinates|canonically conjugate variables]].

First introduced in 1927 by German physicist [[Werner Heisenberg]],<ref name=":0">{{Cite journal |last=Heisenberg |first=W. |orig-date=1927-03-01 |title=Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik |url=https://doi.org/10.1007/BF01397280 |journal=Zeitschrift für Physik |date=1927 |language=de |volume=43 |issue=3 |pages=172–198 |bibcode=1927ZPhy...43..172H |doi=10.1007/BF01397280 |issn=0044-3328 |s2cid=122763326 }}{{Cite journal |last=Heisenberg |first=W |year=1983 |orig-date=1927 |title=The actual content of quantum theoretical kinematics and mechanics |url=https://ntrs.nasa.gov/citations/19840008978 |journal=No. NAS 1.15: 77379. 1983. |volume=43 |issue=3–4 |page=172 |bibcode=1983ZhPhy..43..172H |quote=English translation of Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik |access-date=2023-08-28 |archive-date=2023-09-02 |archive-url=https://web.archive.org/web/20230902112403/https://ntrs.nasa.gov/citations/19840008978 |url-status=live }}</ref><ref>Werner Heisenberg (1989), ''Encounters with Einstein and Other Essays on People, Places and Particles'',  [[Princeton University Press]], p. 53. {{ISBN?}}</ref><ref>{{cite book | doi=10.1515/9781400889167 | title=The Tests of Time | year=2003 | isbn=978-1400889167 | editor-last1=Dolling | editor-last2=Gianelli | editor-last3=Statile | editor-first1=Lisa M. | editor-first2=Arthur F. | editor-first3=Glenn N. }}</ref><ref>Kumar, Manjit. ''Quantum: Einstein, Bohr, and the great debate about the nature of reality.'' 1st American ed., 2008. Chap. 10, Note 37. {{ISBN?}}</ref> the formal inequality relating the [[standard deviation]] of position ''σ<sub>x</sub>'' and the standard deviation of momentum ''σ<sub>p</sub>'' was derived by [[Earle Hesse Kennard]]<ref name="Kennard">{{Citation |first=E. H. |last=Kennard |title=Zur Quantenmechanik einfacher Bewegungstypen |language=de|journal=Zeitschrift für Physik |volume=44 |issue=4–5 |year=1927 |pages=326–352 |doi=10.1007/BF01391200 |postscript=. |bibcode = 1927ZPhy...44..326K |s2cid=121626384 }}</ref> later that year and by [[Hermann Weyl]]<ref name="Weyl1928">{{Cite book |last=Weyl |first=H. |title=Gruppentheorie und Quantenmechanik |lang=de |year=1928 |publisher=Hirzel |location=Leipzig}}{{page?|date=February 2024}}</ref> in 1928:
{{Equation box 1
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|equation = <math> \sigma_{x}\sigma_{p} \geq \frac{\hbar}{2}</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where <math>\hbar = \frac{h}{2\pi}</math> is the [[reduced Planck constant]].

The quintessentially quantum mechanical uncertainty principle comes in many forms other than position–momentum. The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time. The basic principle has been extended in numerous directions; it must be considered in many kinds of fundamental physical measurements.

==Position–momentum==
{{Main article|Introduction to quantum mechanics}}

[[File:Sequential superposition of plane waves.gif|360px|right|thumb|The superposition of several plane waves to form a wave packet. This wave packet becomes increasingly localized with the addition of many waves. The Fourier transform is a mathematical operation that separates a wave packet into its individual plane waves. The waves shown here are [[real function|real]] for illustrative purposes only; in quantum mechanics the wave function is generally [[complex function|complex]].]]

It is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic<ref>{{cite journal | last1=Jaeger|first1=Gregg|title=What in the (quantum) world is macroscopic?|journal=American Journal of Physics|date=September 2014 | volume=82|issue=9|pages=896–905|doi=10.1119/1.4878358|bibcode = 2014AmJPh..82..896J }}</ref> scales that humans experience. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The [[Schrödinger equation|wave mechanics]] picture of the uncertainty principle is more visually intuitive, but the more abstract [[matrix mechanics]] picture formulates it in a way that generalizes more easily.

Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding [[orthonormal]] [[basis (linear algebra)|bases]] in [[Hilbert space]] are [[Fourier transforms]] of one another (i.e., position and momentum are [[conjugate variables]]). A nonzero function and its Fourier transform cannot both be sharply localized at the same time.<ref>See Appendix B in {{citation |title=Why photons cannot be sharply localized |first1=Iwo |last1=Bialynicki-Birula |first2=Zofia |last2=Bialynicka-Birula |journal=Physical Review A |date=2009 |volume=79 |issue=3 |pages=7–8|doi=10.1103/PhysRevA.79.032112 |arxiv=0903.3712 |bibcode=2009PhRvA..79c2112B |s2cid=55632217 }}</ref> A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a [[Dirac delta function|sharp spike]] at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of the particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the [[Matter wave|de Broglie relation]] {{math|''p'' {{=}} ''ħk''}}, where {{mvar|k}} is the [[wavenumber]].

In [[matrix mechanics]], the [[mathematical formulation of quantum mechanics#Postulates of quantum mechanics|mathematical formulation of quantum mechanics]], any pair of non-[[commutator|commuting]] [[self-adjoint operator]]s representing [[observable]]s are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable {{mvar|A}} is performed, then the system is in a particular eigenstate {{mvar|Ψ}} of that observable. However, the particular eigenstate of the observable {{mvar|A}} need not be an eigenstate of another observable {{mvar|B}}: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.<ref>{{Citation|author1=Claude Cohen-Tannoudji | author2=Bernard Diu | author3=Franck Laloë |title=Quantum mechanics|year=1996|publisher=Wiley|location=Wiley-Interscience | isbn=978-0-471-56952-7 | pages=231–233}}</ref>

===Visualization===
The uncertainty principle can be visualized using the position- and momentum-space wavefunctions for one spinless particle with mass in one dimension.

The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentum-space wavefunction is less localized so the possible momentum components the particle could have are more widespread. Conversely, the more localized the momentum-space wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly the less localized the position-space wavefunction, so the position coordinates the particle could occupy are more widespread. These wavefunctions are [[Fourier transform]]s of each other: mathematically, the uncertainty principle expresses the relationship between conjugate variables in the transform.

[[File:Quantum mechanics travelling wavefunctions wavelength.svg|center|thumb|502px|Position ''x'' and momentum ''p'' wavefunctions corresponding to quantum particles. The colour opacity of the particles corresponds to the [[probability density]] of finding the particle with position ''x'' or momentum component ''p''.<br/>

'''Top:''' If wavelength ''λ'' is unknown, so are momentum ''p'', wave-vector ''k'' and energy ''E'' (de Broglie relations). As the particle is more localized in position space, Δ''x'' is smaller than for Δ''p<sub>x</sub>''.<br/>

'''Bottom:''' If ''λ'' is known, so are ''p'', ''k'', and ''E''. As the particle is more localized in momentum space, Δ''p'' is smaller than for Δ''x''.]]

{{Clear}}

===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.

===Proof of the Kennard inequality using wave mechanics===

We are interested in the [[variance]]s of position and momentum, defined as
<math display="block">\sigma_x^2 = \int_{-\infty}^\infty x^2 \cdot |\psi(x)|^2 \, dx - \left( \int_{-\infty}^\infty x \cdot |\psi(x)|^2 \, dx \right)^2</math>
<math display="block">\sigma_p^2 = \int_{-\infty}^\infty p^2 \cdot |\varphi(p)|^2 \, dp - \left( \int_{-\infty}^\infty p \cdot |\varphi(p)|^2 \, dp \right)^2~.</math>

[[Without loss of generality]], we will assume that the [[expected value|means]] vanish, which just amounts to a shift of the origin of our coordinates. (A more general proof that does not make this assumption is given below.) This gives us the simpler form
<math display="block">\sigma_x^2 = \int_{-\infty}^\infty x^2 \cdot |\psi(x)|^2 \, dx</math>
<math display="block">\sigma_p^2 = \int_{-\infty}^\infty p^2 \cdot |\varphi(p)|^2 \, dp~.</math>

The function <math>f(x) = x \cdot \psi(x)</math> can be interpreted as a [[vector space|vector]] in a [[function space]]. We can define an [[inner product]] for a pair of functions ''u''(''x'') and ''v''(''x'') in this vector space:
<math display="block">\langle u \mid v \rangle = \int_{-\infty}^\infty u^*(x) \cdot v(x) \, dx,</math>
where the asterisk denotes the [[complex conjugate]].

With this inner product defined, we note that the variance for position can be written as
<math display="block">\sigma_x^2 = \int_{-\infty}^\infty |f(x)|^2 \, dx = \langle f \mid f \rangle ~.</math>

We can repeat this for momentum by interpreting the function <math>\tilde{g}(p)=p \cdot \varphi(p)</math> as a vector, but we can also take advantage of the fact that <math>\psi(x)</math> and <math>\varphi(p)</math> are Fourier transforms of each other. We evaluate the inverse Fourier transform through [[integration by parts]]:
<math display="block">\begin{align}
g(x) &= \frac{1}{\sqrt{2 \pi \hbar}} \cdot \int_{-\infty}^\infty \tilde{g}(p) \cdot e^{ipx/\hbar} \, dp \\
&= \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^\infty p \cdot \varphi(p) \cdot e^{ipx/\hbar} \, dp \\
&= \frac{1}{2 \pi \hbar} \int_{-\infty}^\infty \left[ p \cdot \int_{-\infty}^\infty \psi(\chi) e^{-ip\chi/\hbar} \, d\chi \right] \cdot e^{ipx/\hbar} \, dp \\
&= \frac{i}{2 \pi} \int_{-\infty}^\infty \left[ \cancel{ \left. \psi(\chi) e^{-ip\chi/\hbar} \right|_{-\infty}^\infty } - \int_{-\infty}^\infty \frac{d\psi(\chi)}{d\chi} e^{-ip\chi/\hbar} \, d\chi \right] \cdot e^{ipx/\hbar} \, dp \\
&= -i \int_{-\infty}^\infty \frac{d\psi(\chi)}{d\chi} \left[ \frac{1}{2 \pi}\int_{-\infty}^\infty \, e^{ip(x - \chi)/\hbar} \, dp \right]\, d\chi\\
&= -i \int_{-\infty}^\infty \frac{d\psi(\chi)}{d\chi} \left[ \delta\left(\frac{x - \chi }{\hbar}\right) \right]\, d\chi\\
&= -i \hbar \int_{-\infty}^\infty \frac{d\psi(\chi)}{d\chi} \left[ \delta\left(x - \chi \right) \right]\, d\chi\\
&= -i \hbar \frac{d\psi(x)}{dx} \\
&= \left( -i \hbar \frac{d}{dx} \right) \cdot \psi(x) ,
\end{align}</math>
where <math>v=\frac{\hbar}{-ip}e^{-ip\chi/\hbar}</math> in the integration by parts, the cancelled term vanishes because the wave function vanishes at both infinities and <math>|e^{-ip\chi/\hbar}|=1</math>, and then use the [[Dirac delta function#History|Dirac delta function]] which is valid because <math>\dfrac{d\psi(\chi)}{d\chi}</math> does not depend on ''p'' .

The term <math display="inline">-i \hbar \frac{d}{dx}</math> is called the [[momentum operator]] in position space. Applying [[Plancherel theorem|Plancherel's theorem]], we see that the variance for momentum can be written as
<math display="block">\sigma_p^2 = \int_{-\infty}^\infty |\tilde{g}(p)|^2 \, dp = \int_{-\infty}^\infty |g(x)|^2 \, dx = \langle g \mid g \rangle.</math>

The [[Cauchy–Schwarz inequality]] asserts that
<math display="block">\sigma_x^2 \sigma_p^2 = \langle f \mid f \rangle \cdot \langle g \mid g \rangle \ge |\langle f \mid g \rangle|^2 ~.</math>

The [[modulus squared]] of any complex number ''z'' can be expressed as
<math display="block">|z|^{2} = \Big(\text{Re}(z)\Big)^{2}+\Big(\text{Im}(z)\Big)^{2} \geq \Big(\text{Im}(z)\Big)^{2} = \left(\frac{z-z^{\ast}}{2i}\right)^{2}. </math>
we let <math>z=\langle f|g\rangle</math> and <math>z^{*}=\langle g\mid f\rangle</math> and substitute these into the equation above to get
<math display="block">|\langle f\mid g\rangle|^2 \geq \left(\frac{\langle f\mid g\rangle-\langle g \mid f \rangle}{2i}\right)^2 ~.</math>

All that remains is to evaluate these inner products.

<math display="block">\begin{align}
\langle f\mid g\rangle-\langle g\mid f\rangle  &= \int_{-\infty}^\infty \psi^*(x) \, x \cdot \left(-i \hbar \frac{d}{dx}\right) \, \psi(x) \, dx - \int_{-\infty}^\infty \psi^*(x) \, \left(-i \hbar \frac{d}{dx}\right) \cdot x \, \psi(x) \, dx \\
&= i \hbar \cdot \int_{-\infty}^\infty \psi^*(x) \left[ \left(-x \cdot \frac{d\psi(x)}{dx}\right) + \frac{d(x \psi(x))}{dx} \right] \, dx \\
&= i \hbar \cdot \int_{-\infty}^\infty \psi^*(x) \left[ \left(-x \cdot \frac{d\psi(x)}{dx}\right) + \psi(x) + \left(x \cdot \frac{d\psi(x)}{dx}\right)\right] \, dx \\
&= i \hbar \cdot \int_{-\infty}^\infty \psi^*(x) \psi(x) \, dx \\
&= i \hbar \cdot \int_{-\infty}^\infty |\psi(x)|^2 \, dx \\
&= i \hbar
\end{align}</math>

Plugging this into the above inequalities, we get
<math display="block">\sigma_x^2 \sigma_p^2 \ge |\langle f \mid g \rangle|^2 \ge \left(\frac{\langle f\mid g\rangle-\langle g\mid f\rangle}{2i}\right)^2 = \left(\frac{i \hbar}{2 i}\right)^2 = \frac{\hbar^2}{4}</math>
and taking the square root
<math display="block">\sigma_x \sigma_p \ge \frac{\hbar}{2}~.</math>

with equality if and only if ''p'' and ''x'' are linearly dependent. Note that the only ''physics'' involved in this proof was that <math>\psi(x)</math> and <math>\varphi(p)</math> are wave functions for position and momentum, which are Fourier transforms of each other. A similar result would hold for ''any'' pair of conjugate variables.

===Matrix mechanics interpretation===
{{Main article|Matrix mechanics}}
In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators.<ref name="L&L"/> When considering pairs of observables, an important quantity is the ''[[commutator]]''. For a pair of operators {{mvar|Â}} and <math>\hat{B}</math>, one defines their commutator as
<math display="block">[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}.</math>
In the case of position and momentum, the commutator is the [[canonical commutation relation]]
<math display="block">[\hat{x},\hat{p}]=i \hbar.</math>

The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum [[eigenstate]]s. Let <math>|\psi\rangle</math> be a right eigenstate of position with a constant eigenvalue {{math|''x''<sub>0</sub>}}. By definition, this means that <math>\hat{x}|\psi\rangle = x_0 |\psi\rangle.</math> Applying the commutator to <math>|\psi\rangle</math> yields
<math display="block">[\hat{x},\hat{p}] | \psi \rangle = (\hat{x}\hat{p}-\hat{p}\hat{x}) | \psi \rangle = (\hat{x} - x_0 \hat{I}) \hat{p} \, | \psi \rangle = i \hbar | \psi \rangle,</math>
where {{mvar|Î}} is the [[identity matrix|identity operator]].

Suppose, for the sake of [[proof by contradiction]], that <math>|\psi\rangle</math> is also a right eigenstate of momentum, with constant eigenvalue {{mvar|''p''<sub>0</sub>}}. If this were true, then one could write
<math display="block">(\hat{x} - x_0 \hat{I}) \hat{p} \, | \psi \rangle = (\hat{x} - x_0 \hat{I}) p_0 \, | \psi \rangle = (x_0 \hat{I} - x_0 \hat{I}) p_0 \, | \psi \rangle=0.</math>
On the other hand, the above canonical commutation relation requires that
<math display="block">[\hat{x},\hat{p}] | \psi \rangle=i \hbar | \psi \rangle \ne 0.</math>
This implies that no quantum state can simultaneously be both a position and a momentum eigenstate.

When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is ''not'' a momentum eigenstate, however, but rather it can be represented as a sum of multiple momentum basis eigenstates. In other words, the momentum must be less precise. This precision may be quantified by the standard deviations,
<math display="block">\sigma_x=\sqrt{\langle \hat{x}^2 \rangle-\langle \hat{x}\rangle^2}</math>
<math display="block">\sigma_p=\sqrt{\langle \hat{p}^2 \rangle-\langle \hat{p}\rangle^2}.</math>

As in the wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle.

===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]].

=== Quantum harmonic oscillators with Gaussian initial condition ===
{{multiple image
| align = right
| direction = vertical
| footer =
Position (blue) and momentum (red) probability densities for an initial Gaussian distribution. From top to bottom, the animations show the cases {{nowrap|1=Ω = ''ω''}}, {{nowrap|1=Ω = 2''ω''}}, and {{nowrap|1=Ω = ''ω''/2}}. Note the tradeoff between the widths of the distributions.
| width1 = 360
| image1 = Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_balanced.gif
| width2 = 360
| image2 = Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_narrow.gif
| width3 = 360
| image3 = Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_wide.gif
}}

In a quantum harmonic oscillator of characteristic angular frequency ''ω'', place a state that is offset from the bottom of the potential by some displacement ''x''<sub>0</sub> as
<math display="block">\psi(x)=\left(\frac{m \Omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \Omega (x-x_0)^2}{2\hbar}\right)},</math>
where Ω describes the width of the initial state but need not be the same as ''ω''. Through integration over the [[Propagator#Basic examples: propagator of free particle and harmonic oscillator|propagator]], we can solve for the {{Not a typo|full time}}-dependent solution. After many cancelations, the probability densities reduce to
<math display="block">|\Psi(x,t)|^2 \sim \mathcal{N}\left( x_0 \cos{(\omega t)} , \frac{\hbar}{2 m \Omega} \left( \cos^2(\omega t) + \frac{\Omega^2}{\omega^2} \sin^2{(\omega t)} \right)\right)</math>
<math display="block">|\Phi(p,t)|^2 \sim \mathcal{N}\left( -m x_0 \omega \sin(\omega t), \frac{\hbar m \Omega}{2} \left( \cos^2{(\omega t)} + \frac{\omega^2}{\Omega^2} \sin^2{(\omega t)} \right)\right),</math>
where we have used the notation <math>\mathcal{N}(\mu, \sigma^2)</math> to denote a normal distribution of mean ''μ'' and variance ''σ''<sup>2</sup>. Copying the variances above and applying [[list of trigonometric identities|trigonometric identities]], we can write the product of the standard deviations as
<math display="block">\begin{align}
\sigma_x \sigma_p&=\frac{\hbar}{2}\sqrt{\left( \cos^2{(\omega t)} + \frac{\Omega^2}{\omega^2} \sin^2{(\omega t)} \right)\left( \cos^2{(\omega t)} + \frac{\omega^2}{\Omega^2} \sin^2{(\omega t)} \right)} \\
&= \frac{\hbar}{4}\sqrt{3+\frac{1}{2}\left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-\left(\frac{1}{2}\left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-1\right) \cos{(4 \omega t)}}
\end{align}</math>

From the relations
<math display="block">\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2} \ge 2, \quad |\cos(4 \omega t)| \le 1,</math>
we can conclude the following (the right most equality holds only when {{nowrap|1=Ω = ''ω''}}):
<math display="block">\sigma_x \sigma_p \ge \frac{\hbar}{4}\sqrt{3+\frac{1}{2} \left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-\left(\frac{1}{2} \left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-1\right)} = \frac{\hbar}{2}. </math>

===Coherent states===
{{Main article|Coherent state}}
A coherent state is a right eigenstate of the [[annihilation operator]],
<math display="block">\hat{a}|\alpha\rangle=\alpha|\alpha\rangle,</math>
which may be represented in terms of [[Fock state]]s as
<math display="block">|\alpha\rangle =e^{-{|\alpha|^2\over2}} \sum_{n=0}^\infty {\alpha^n \over \sqrt{n!}}|n\rangle</math>

In the picture where the coherent state is a massive particle in a quantum harmonic oscillator, the position and momentum operators may be expressed in terms of the annihilation operators in the same formulas above and used to calculate the variances,
<math display="block">\sigma_x^2 = \frac{\hbar}{2 m \omega},</math>
<math display="block">\sigma_p^2 = \frac{\hbar m \omega}{2}.</math>
Therefore, every coherent state saturates the Kennard bound
<math display="block">\sigma_x \sigma_p = \sqrt{\frac{\hbar}{2 m \omega}} \, \sqrt{\frac{\hbar m \omega}{2}} = \frac{\hbar}{2}. </math>
with position and momentum each contributing an amount <math display="inline">\sqrt{\hbar/2}</math> in a "balanced" way. Moreover, every [[squeezed coherent state]] also saturates the Kennard bound although the individual contributions of position and momentum need not be balanced in general.

===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>

===Constant momentum===
{{Main article|Wave packet}}
[[File:Guassian Dispersion.gif|360 px|thumb|right|Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space]]
Assume a particle initially has a [[momentum space]] wave function described by a normal distribution around some constant momentum ''p''<sub>0</sub> according to
<math display="block">\varphi(p) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \exp\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}\right),</math>
where we have introduced a reference scale <math display="inline">x_0=\sqrt{\hbar/m\omega_0}</math>, with <math>\omega_0>0</math> describing the width of the distribution—cf. [[nondimensionalization]]. If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are
<math display="block">\Phi(p,t) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \exp\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}-\frac{ip^2 t}{2m\hbar}\right),</math>
<math display="block">\Psi(x,t) = \left(\frac{1}{x_0 \sqrt{\pi}} \right)^{1/2} \frac{e^{-x_0^2 p_0^2 /2\hbar^2}}{\sqrt{1+i\omega_0 t}} \, \exp\left(-\frac{(x-ix_0^2 p_0/\hbar)^2}{2x_0^2 (1+i\omega_0 t)}\right).</math>

Since <math> \langle p(t) \rangle = p_0</math> and <math>\sigma_p(t) = \hbar /(\sqrt{2}x_0)</math>, this can be interpreted as a particle moving along with constant momentum at arbitrarily high precision. On the other hand, the standard deviation of the position is
<math display="block">\sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2}</math>
such that the uncertainty product can only increase with time as
<math display="block">\sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2}</math>

==Mathematical formalism==
Starting with Kennard's derivation of position-momentum uncertainty, [[Howard Percy Robertson]] developed<ref name="Robertson1929">{{Citation|last=Robertson|first=H. P.|title=The Uncertainty Principle|journal=Phys. Rev. | year=1929|volume=34|issue=1|pages=163–164|bibcode = 1929PhRv...34..163R |doi = 10.1103/PhysRev.34.163 }}</ref><ref name=Sen2014/> a formulation for arbitrary [[Self-adjoint operator|Hermitian operator]]s <math>\hat{\mathcal{O}}</math> expressed in terms of their standard deviation
<math display="block">\sigma_{\mathcal{O}} = \sqrt{\langle \hat{\mathcal{O}}^2 \rangle-\langle \hat{\mathcal{O}}\rangle^2},</math>
where the brackets <math>\langle\hat{\mathcal{O}}\rangle</math> indicate an [[expectation value (quantum mechanics)|expectation value]] of the observable represented by operator <math>\hat{\mathcal{O}}</math>. For a pair of operators <math>\hat{A}</math> and <math>\hat{B}</math>, define their commutator as
<math display="block">[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A},</math>
and the Robertson uncertainty relation is given by<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 242–243 | bibcode = 2013qtm..book.....H }}</ref>
<math display="block">\sigma_A \sigma_B \geq \left| \frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle \right| = \frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle \right|.</math>

[[Erwin Schrödinger]]<ref>Schrödinger, E., Zum Heisenbergschen Unschärfeprinzip, Berliner Berichte, 1930, pp. 296–303.</ref> showed how to allow for correlation between the operators, giving a stronger inequality, known as the '''Robertson–Schrödinger uncertainty relation''',<ref name="Schrodinger1930">{{Citation | last = Schrödinger |first = E. | title = Zum Heisenbergschen Unschärfeprinzip | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse | volume = 14 | pages = 296–303 | year = 1930}}</ref><ref name=Sen2014/>

{{Equation box 1
|indent =:
|equation = <math>\sigma_A^2\sigma_B^2 \geq \left| \frac{1}{2}\langle\{\hat{A}, \hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle \right|^2+ \left|\frac{1}{2i} \langle[ \hat{A}, \hat{B}] \rangle\right|^2,</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where the anticommutator, <math>\{\hat{A},\hat{B}\}=\hat{A}\hat{B}+\hat{B}\hat{A}</math> is used.

{{math proof
|title=Proof of the [[Erwin Schrödinger|Schrödinger]] uncertainty relation
|proof=
The derivation shown here incorporates and builds off of those shown in Robertson,<ref name="Robertson1929" /> Schrödinger<ref name="Schrodinger1930" /> and standard textbooks such as Griffiths.<ref name="GriffithsSchroeter2018">{{Cite book |last1=Griffiths |first1=David J. |url=https://www.cambridge.org/highereducation/product/9781316995433/book |title=Introduction to Quantum Mechanics |last2=Schroeter |first2=Darrell F. |year=2018 |publisher=Cambridge University Press |isbn=978-1-316-99543-3 |edition=3rd |doi=10.1017/9781316995433 |bibcode=2018iqm..book.....G |access-date=2024-01-27 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223160131/https://www.cambridge.org/highereducation/books/introduction-to-quantum-mechanics/990799CA07A83FC5312402AF6860311E#overview |url-status=live }}</ref>{{rp|138}} For any Hermitian operator <math>\hat{A}</math>, based upon the definition of [[variance]], we have
<math display="block"> \sigma_A^2 = \langle(\hat{A}-\langle \hat{A} \rangle)\Psi|(\hat{A}-\langle \hat{A} \rangle)\Psi\rangle. </math>
we let <math>|f\rangle=|(\hat{A}-\langle \hat{A} \rangle)\Psi\rangle </math> and thus
<math display="block"> \sigma_A^2 = \langle f\mid f\rangle\, .</math>

Similarly, for any other Hermitian operator <math> \hat{B} </math> in the same state
<math display="block"> \sigma_B^2 = \langle(\hat{B}-\langle \hat{B} \rangle)\Psi|(\hat{B}-\langle \hat{B} \rangle)\Psi\rangle = \langle g\mid g\rangle </math>
for <math> |g\rangle=|(\hat{B}-\langle \hat{B} \rangle)\Psi \rangle.</math>

The product of the two deviations can thus be expressed as

{{NumBlk|:|<math> \sigma_A^2\sigma_B^2 = \langle f\mid f\rangle\langle g\mid g\rangle. </math>|{{EquationRef|1}}}}

In order to relate the two vectors <math>|f\rangle</math> and <math>|g\rangle</math>, we use the [[Cauchy–Schwarz inequality]]<ref name="Riley2006">{{Citation | last = Riley | first = K. F. | author2 = M. P. Hobson and S. J. Bence | title = Mathematical Methods for Physics and Engineering | publisher = Cambridge | year = 2006 | page = 246 }}{{ISBN?}}</ref> which is defined as
<math display="block">\langle f\mid f\rangle\langle g\mid g\rangle \geq |\langle f\mid g\rangle|^2, </math>
and thus Equation ({{EquationNote|1}}) can be written as

{{NumBlk|:|<math>\sigma_A^2\sigma_B^2 \geq |\langle f\mid g\rangle|^2.</math>|{{EquationRef|2}}}}

Since <math> \langle f\mid g\rangle</math> is in general a complex number, we use the fact that the modulus squared of any complex number <math>z</math> is defined as <math>|z|^2=zz^{*}</math>, where <math>z^{*}</math> is the complex conjugate of <math>z</math>. The modulus squared can also be expressed as

{{NumBlk|:|<math> |z|^2 = \Big(\operatorname{Re}(z)\Big)^2+\Big(\operatorname{Im}(z)\Big)^2 = \Big(\frac{z+z^\ast}{2}\Big)^2 +\Big(\frac{z-z^\ast}{2i}\Big)^2. </math>|{{EquationRef|3}}}}

we let <math>z=\langle f\mid g\rangle</math> and <math>z^{*}=\langle g \mid f \rangle </math> and substitute these into the equation above to get

{{NumBlk|:|<math>|\langle f\mid g\rangle|^2 = \bigg(\frac{\langle f\mid g\rangle+\langle g\mid f\rangle}{2}\bigg)^2 + \bigg(\frac{\langle f\mid g\rangle-\langle g\mid f\rangle}{2i}\bigg)^2 </math>|{{EquationRef|4}}}}

The inner product <math>\langle f\mid g\rangle </math> is written out explicitly as
<math display="block">\langle f\mid g\rangle = \langle(\hat{A}-\langle \hat{A} \rangle)\Psi|(\hat{B}-\langle \hat{B} \rangle)\Psi\rangle,</math>
and using the fact that <math>\hat{A}</math> and <math>\hat{B}</math> are Hermitian operators, we find
<math display="block">
\begin{align}
\langle f\mid g\rangle & = \langle\Psi|(\hat{A}-\langle \hat{A}\rangle)(\hat{B}-\langle \hat{B}\rangle)\Psi\rangle \\[4pt]
& = \langle\Psi\mid(\hat{A}\hat{B}-\hat{A}\langle \hat{B}\rangle - \hat{B}\langle \hat{A}\rangle + \langle \hat{A}\rangle\langle \hat{B}\rangle)\Psi\rangle \\[4pt]
& = \langle\Psi\mid\hat{A}\hat{B}\Psi\rangle-\langle\Psi\mid\hat{A}\langle \hat{B}\rangle\Psi\rangle
-\langle\Psi\mid\hat{B}\langle \hat{A}\rangle\Psi\rangle+\langle\Psi\mid\langle \hat{A}\rangle\langle \hat{B}\rangle\Psi\rangle \\[4pt]
& =\langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle+\langle \hat{A}\rangle\langle \hat{B}\rangle \\[4pt]
& =\langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle.
\end{align}
</math>

Similarly it can be shown that <math>\langle g\mid f\rangle = \langle \hat{B}\hat{A}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle.</math>

Thus, we have
<math display="block">
\langle f\mid g\rangle-\langle g\mid f\rangle = \langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle-\langle \hat{B}\hat{A}\rangle+\langle \hat{A}\rangle\langle \hat{B}\rangle = \langle [\hat{A},\hat{B}]\rangle
</math>
and
<math display="block">\langle f\mid g\rangle+\langle g\mid f\rangle = \langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle+\langle \hat{B}\hat{A}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle = \langle \{\hat{A},\hat{B}\}\rangle -2\langle \hat{A}\rangle\langle \hat{B}\rangle. </math>

We now substitute the above two equations above back into Eq. ({{EquationNote|4}}) and get
<math display="block">|\langle f\mid g\rangle|^2=\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle\Big)^2 + \Big(\frac{1}{2i} \langle[\hat{A},\hat{B}]\rangle\Big)^{2}\, .</math>

Substituting the above into Equation ({{EquationNote|2}}) we get the Schrödinger uncertainty relation
<math display="block">\sigma_A\sigma_B \geq \sqrt{\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle\Big)^2 + \Big(\frac{1}{2i} \langle[\hat{A},\hat{B}]\rangle\Big)^2}.</math>

This proof has an issue<ref>{{Citation|last=Davidson|first=E. R.|title=On Derivations of the Uncertainty Principle|journal=J. Chem. Phys.|volume=42|year=1965|doi=10.1063/1.1696139|bibcode = 1965JChPh..42.1461D|issue=4|pages=1461–1462 }}</ref> related to the domains of the operators involved. For the proof to make sense, the vector <math> \hat{B} |\Psi \rangle</math> has to be in the domain of the [[unbounded operator]] <math> \hat{A}</math>, which is not always the case. In fact, the Robertson uncertainty relation is false if <math>\hat{A}</math> is an angle variable and <math>\hat{B}</math> is the derivative with respect to this variable. In this example, the commutator is a nonzero constant—just as in the Heisenberg uncertainty relation—and yet there are states where the product of the uncertainties is zero.<ref name="Hall2013"/> (See the counterexample section below.) This issue can be overcome by using a [[variational method]] for the proof,<ref name="Jackiw">{{Citation|last=Jackiw| first=Roman|title=Minimum Uncertainty Product, Number-Phase Uncertainty Product, and Coherent States|journal=J. Math. Phys.|volume=9|year=1968|doi=10.1063/1.1664585|bibcode = 1968JMP.....9..339J|issue=3|pages=339–346 }}</ref><ref name="CarruthersNieto">{{Citation|first1=P. |last1=Carruthers|last2= Nieto|first2=M. M.|title=Phase and Angle Variables in Quantum Mechanics|journal=Rev. Mod. Phys.|volume=40|year=1968|doi=10.1103/RevModPhys.40.411|bibcode = 1968RvMP...40..411C|issue=2|pages=411–440 }}</ref> or by working with an exponentiated version of the canonical commutation relations.<ref name="Hall2013"/>

Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operators <math>\hat{A}</math> and <math>\hat{B}</math> are [[Self-adjoint operator#Self-adjoint operators|self-adjoint operators]]. It suffices to assume that they are merely [[Self-adjoint operator#Symmetric operators|symmetric operators]]. (The distinction between these two notions is generally glossed over in the physics literature, where the term ''Hermitian'' is used for either or both classes of operators. See Chapter 9 of Hall's book<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | bibcode = 2013qtm..book.....H }}</ref> for a detailed discussion of this important but technical distinction.)
}}

===Phase space===
In the [[phase space formulation]] of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a [[Wigner quasi-probability distribution|Wigner function]] <math>W(x,p)</math> with [[Moyal product|star product]] ★ and a function ''f'', the following is generally true:<ref>{{Cite journal | last1 = Curtright | first1 = T. |last2= Zachos | first2= C. | title = Negative Probability and Uncertainty Relations| journal = Modern Physics Letters A | volume = 16 | issue = 37 | pages = 2381–2385 | doi = 10.1142/S021773230100576X | year = 2001 |arxiv = hep-th/0105226 |bibcode = 2001MPLA...16.2381C | s2cid = 119669313 }}</ref>
<math display="block">\langle f^* \star f \rangle =\int (f^* \star f) \, W(x,p) \, dx \, dp \ge 0 ~.</math>

Choosing <math>f = a + bx + cp</math>, we arrive at
<math display="block">\langle f^* \star f \rangle =\begin{bmatrix}a^* & b^* & c^* \end{bmatrix}\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix}\begin{bmatrix}a \\ b \\ c\end{bmatrix} \ge 0 ~.</math>

Since this positivity condition is true for ''all'' ''a'', ''b'', and ''c'', it follows that all the eigenvalues of the matrix are non-negative.

The non-negative eigenvalues then imply a corresponding non-negativity condition on the [[determinant]],
<math display="block">\det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix}
= \det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x^2 \rangle & \left\langle xp + \frac{i\hbar}{2} \right\rangle \\ \langle p \rangle & \left\langle xp - \frac{i\hbar}{2} \right\rangle & \langle p^2 \rangle \end{bmatrix}
\ge 0~,</math>
or, explicitly, after algebraic manipulation,
<math display="block">\sigma_x^2 \sigma_p^2 = \left( \langle x^2 \rangle - \langle x \rangle^2 \right)\left( \langle p^2 \rangle - \langle p \rangle^2 \right)\ge \left( \langle xp \rangle - \langle x \rangle \langle p \rangle \right)^2 + \frac{\hbar^2}{4} ~.</math>

===Examples===

Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below.
* '''Position–linear momentum uncertainty relation''': for the position and linear momentum operators, the canonical commutation relation <math>[\hat{x}, \hat{p}] = i\hbar</math> implies the Kennard inequality from above: <math display="block">\sigma_x \sigma_p \geq \frac{\hbar}{2}.</math>
* '''Angular momentum uncertainty relation''': For two orthogonal components of the [[angular momentum|total angular momentum]] operator of an object: <math display="block">\sigma_{J_i} \sigma_{J_j} \geq \frac{\hbar}{2} \big|\langle J_k\rangle\big|,</math> where ''i'', ''j'', ''k'' are distinct, and ''J''<sub>''i''</sub> denotes angular momentum along the ''x''<sub>''i''</sub> axis. This relation implies that unless all three components vanish together, only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field. Moreover, for <math>[J_x, J_y] = i \hbar \varepsilon_{xyz} J_z</math>, a choice <math>\hat{A} = J_x</math>, <math>\hat{B} = J_y</math>, in angular momentum multiplets, ''ψ'' = |''j'', ''m''⟩, bounds the [[Casimir invariant]] (angular momentum squared, <math>\langle J_x^2+ J_y^2 + J_z^2 \rangle</math>) from below and thus yields useful constraints such as {{nobr|''j''(''j'' + 1) ≥ ''m''(''m'' + 1)}}, and hence ''j'' ≥ ''m'', among others.

* For the number of electrons in a [[superconductor]] and the [[Phase factor|phase]] of its [[Ginzburg–Landau theory|Ginzburg–Landau order parameter]]<ref>{{Citation |last=Likharev |first=K. K. |author2=A. B. Zorin |title=Theory of Bloch-Wave Oscillations in Small Josephson Junctions |journal=J. Low Temp. Phys. |volume=59 |issue=3/4 |pages=347–382 |year=1985 |doi=10.1007/BF00683782 |bibcode=1985JLTP...59..347L|s2cid=120813342 }}</ref><ref>{{Citation |first=P. W. |last=Anderson |editor-last=Caianiello |editor-first=E. R. |contribution=Special Effects in Superconductivity |title=Lectures on the Many-Body Problem, Vol. 2 |year=1964 |place=New York |publisher=Academic Press}}</ref> <math display="block"> \Delta N \, \Delta \varphi \geq 1. </math>

===Limitations===
The derivation of the Robertson inequality for operators <math>\hat{A}</math> and <math>\hat{B}</math> requires <math>\hat{A}\hat{B}\psi</math> and <math>\hat{B}\hat{A}\psi</math> to be defined. There are quantum systems where these conditions are not valid.<ref>{{Cite journal |last=Davidson |first=Ernest R. |date=1965-02-15 |title=On Derivations of the Uncertainty Principle |url=https://pubs.aip.org/jcp/article/42/4/1461/208937/On-Derivations-of-the-Uncertainty-Principle |journal=The Journal of Chemical Physics |language=en |volume=42 |issue=4 |pages=1461–1462 |doi=10.1063/1.1696139 |bibcode=1965JChPh..42.1461D |issn=0021-9606 |access-date=2024-01-20 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223160247/https://pubs.aip.org/aip/jcp/article-abstract/42/4/1461/208937/On-Derivations-of-the-Uncertainty-Principle?redirectedFrom=fulltext |url-status=live }}</ref>
One example is a quantum [[particle in a ring|particle on a ring]], where the wave function depends on an angular variable <math>\theta</math> in the interval <math>[0,2\pi]</math>. Define "position" and "momentum" operators <math>\hat{A}</math> and <math>\hat{B}</math> by
<math display="block">\hat{A}\psi(\theta)=\theta\psi(\theta),\quad \theta\in [0,2\pi],</math>
and
<math display="block">\hat{B}\psi=-i\hbar\frac{d\psi}{d\theta},</math>
with periodic boundary conditions on <math>\hat{B}</math>. The definition of <math>\hat{A}</math> depends the <math>\theta</math> range from 0 to <math>2\pi</math>. These operators satisfy the usual commutation relations for position and momentum operators, <math>[\hat{A},\hat{B}]=i\hbar</math>. More precisely, <math>\hat{A}\hat{B}\psi-\hat{B}\hat{A}\psi=i\hbar\psi</math> whenever both <math>\hat{A}\hat{B}\psi</math> and <math>\hat{B}\hat{A}\psi</math> are defined, and the space of such <math>\psi</math> is a dense subspace of the quantum Hilbert space.<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | page = 245 | bibcode = 2013qtm..book.....H }}</ref>

Now let <math>\psi</math> be any of the eigenstates of <math>\hat{B}</math>, which are given by <math>\psi(\theta)=e^{2\pi in\theta}</math>. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator <math>\hat{A}</math> is bounded, since <math>\theta</math> ranges over a bounded interval. Thus, in the state <math>\psi</math>, the uncertainty of <math>B</math> is zero and the uncertainty of <math>A</math> is finite, so that
<math display="block">\sigma_A\sigma_B=0.</math>
The Robertson uncertainty principle does not apply in this case: <math>\psi</math> is not in the domain of the operator <math>\hat{B}\hat{A}</math>, since multiplication by <math>\theta</math> disrupts the periodic boundary conditions imposed on <math>\hat{B}</math>.<ref name="Hall2013">{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 245 | bibcode = 2013qtm..book.....H }}</ref>

For the usual position and momentum operators <math>\hat{X}</math> and <math>\hat{P}</math> on the real line, no such counterexamples can occur. As long as <math>\sigma_x</math> and <math>\sigma_p</math> are defined in the state <math>\psi</math>, the Heisenberg uncertainty principle holds, even if <math>\psi</math> fails to be in the domain of <math>\hat{X}\hat{P}</math> or of <math>\hat{P}\hat{X}</math>.<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 246 | bibcode = 2013qtm..book.....H }}</ref>

===Mixed states===

The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components <math>\varrho_k</math> in any decomposition of the [[density matrix]] given as
<math display="block">
\varrho=\sum_k p_k \varrho_k.
</math>
Here, for the probabilities <math>p_k\ge0</math> and <math>\sum_k p_k=1</math> hold. Then, using the relation
<math display="block">
\sum_k a_k \sum_k b_k \ge \left(\sum_k \sqrt{a_k b_k}\right)^2
</math>
for <math> a_k,b_k\ge 0</math>,
it follows that<ref name="PhysRevResearch21">{{cite journal |last1=Tóth |first1=Géza |last2=Fröwis |first2=Florian |title=Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices |journal=Physical Review Research |date=31 January 2022 |volume=4 |issue=1 |pages=013075 |doi=10.1103/PhysRevResearch.4.013075|arxiv=2109.06893 |bibcode=2022PhRvR...4a3075T |s2cid=237513549 }}</ref>
<math display="block">
\sigma_A^2 \sigma_B^2 \geq \left[\sum_k p_k L(\varrho_k)\right]^2,
</math>
where the function in the bound is defined
<math display="block">
L(\varrho) = \sqrt{\left | \frac{1}{2}\operatorname{tr}(\rho\{A,B\}) - \operatorname{tr}(\rho A)\operatorname{tr}(\rho B)\right |^2 +\left | \frac{1}{2i} \operatorname{tr}(\rho[A,B])\right | ^2}.
</math>
The above relation very often has a bound larger than that of the original Robertson–Schrödinger uncertainty relation. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. The following expression is stronger than the Robertson–Schrödinger uncertainty relation
<math display="block">
\sigma_A^2 \sigma_B^2 \geq \left[\max_{p_k,\varrho_k} \sum_k p_k L(\varrho_k)\right]^2,
</math>
where on the right-hand side there is a concave roof over the decompositions of the density matrix.
The improved relation above is saturated by all single-qubit quantum states.<ref name="PhysRevResearch21" />

With similar arguments, one can derive a relation with a convex roof on the right-hand side<ref name="PhysRevResearch21" />
<math display="block">
\sigma_A^2 F_Q[\varrho,B] \geq 4 \left[\min_{p_k,\Psi_k} \sum_k p_k L(\vert \Psi_k\rangle\langle \Psi_k\vert)\right]^2
</math>
where <math>F_Q[\varrho,B]</math> denotes the [[quantum Fisher information]] and the density matrix is decomposed to pure states as
<math display="block">
\varrho=\sum_k p_k \vert \Psi_k\rangle \langle \Psi_k\vert.
 </math>
The derivation takes advantage of the fact that the [[quantum Fisher information]] is the convex roof of the variance times four.<ref>{{cite journal |last1=Tóth |first1=Géza |last2=Petz |first2=Dénes |title=Extremal properties of the variance and the quantum Fisher information |journal=Physical Review A |date=20 March 2013 |volume=87 |issue=3 |pages=032324 |doi=10.1103/PhysRevA.87.032324|bibcode=2013PhRvA..87c2324T |arxiv=1109.2831 |s2cid=55088553 }}</ref><ref>{{cite arXiv |last1=Yu |first1=Sixia |title=Quantum Fisher Information as the Convex Roof of Variance |date=2013 |eprint=1302.5311|class=quant-ph }}</ref>

A simpler inequality follows without a convex roof<ref>{{cite journal |last1=Fröwis |first1=Florian |last2=Schmied |first2=Roman |last3=Gisin |first3=Nicolas |title=Tighter quantum uncertainty relations following from a general probabilistic bound |journal=Physical Review A |date=2 July 2015 |volume=92 |issue=1 |pages=012102 |doi=10.1103/PhysRevA.92.012102|arxiv=1409.4440 |bibcode=2015PhRvA..92a2102F |s2cid=58912643 }}</ref>
<math display="block">
\sigma_A^2 F_Q[\varrho,B] \geq \vert \langle i[A,B]\rangle\vert^2,
</math>
which is stronger than the Heisenberg uncertainty relation, since for the quantum Fisher information we have
<math display="block">
F_Q[\varrho,B]\le 4 \sigma_B,
</math>
while for pure states the equality holds.

===The Maccone–Pati uncertainty relations===
The Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Lorenzo Maccone and [[Arun K. Pati]] give non-trivial bounds on the sum of the variances for two incompatible observables.<ref>{{cite journal|last1=Maccone|first1=Lorenzo|last2=Pati|first2=Arun K.|title=Stronger Uncertainty Relations for All Incompatible Observables|journal=Physical Review Letters|date=31 December 2014|volume=113| issue=26|page=260401|doi=10.1103/PhysRevLett.113.260401|pmid=25615288|arxiv=1407.0338|bibcode=2014PhRvL.113z0401M|s2cid=21334130 }}</ref> (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref.<ref>{{cite journal |last1=Huang |first1=Yichen |title=Variance-based uncertainty relations |journal=Physical Review A |date=10 August 2012 |volume=86 |issue=2 |page=024101 |doi=10.1103/PhysRevA.86.024101|arxiv=1012.3105 |bibcode=2012PhRvA..86b4101H |s2cid=118507388 }}</ref> due to Yichen Huang.) For two non-commuting observables <math>A</math> and <math>B</math> the first stronger uncertainty relation is given by
<math display="block"> \sigma_{A}^2 + \sigma_{ B}^2 \ge \pm i \langle \Psi\mid [A, B]|\Psi \rangle + \mid \langle \Psi\mid(A \pm i B)\mid{\bar \Psi} \rangle|^2, </math>
where <math> \sigma_{A}^2 = \langle \Psi |A^2 |\Psi \rangle - \langle \Psi \mid A \mid \Psi \rangle^2 </math>, <math> \sigma_{B}^2 = \langle \Psi |B^2 |\Psi \rangle - \langle \Psi \mid B \mid\Psi \rangle^2 </math>, <math>|{\bar \Psi} \rangle </math> is a normalized vector that is orthogonal to the state of the system <math>|\Psi \rangle </math> and one should choose the sign of <math>\pm i \langle \Psi\mid[A, B]\mid\Psi \rangle </math> to make this real quantity a positive number.

The second stronger uncertainty relation is given by
<math display="block"> \sigma_A^2 + \sigma_B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} \mid(A + B)\mid \Psi \rangle|^2 </math>
where <math>| {\bar \Psi}_{A+B} \rangle </math> is a state orthogonal to <math> |\Psi \rangle </math>.
The form of <math>| {\bar \Psi}_{A+B} \rangle </math> implies that the right-hand side of the new uncertainty relation is nonzero unless <math>| \Psi\rangle </math> is an eigenstate of <math>(A + B)</math>. One may note that <math>|\Psi \rangle </math> can be an eigenstate of <math>( A+ B)</math> without being an eigenstate of either <math> A</math> or <math> B </math>. However, when <math> |\Psi \rangle </math> is an eigenstate of one of the two observables the Heisenberg–Schrödinger uncertainty relation becomes trivial. But the lower bound in the new relation is nonzero unless <math> |\Psi \rangle </math> is an eigenstate of both.

== Energy–time ==
{{anchor|Time–energy uncertainty relation}}
An energy–time uncertainty relation like
<math display="block"> \Delta E \Delta t \gtrsim \hbar/2,</math> has a long, controversial history; the meaning of <math>\Delta t</math> and <math>\Delta E</math> varies and different formulations have different arenas of validity.<ref name="Busch2002">{{Cite book |last=Busch |first=Paul |url=http://link.springer.com/10.1007/3-540-45846-8_3 |title=Time in Quantum Mechanics. Lecture Notes in Physics |date=2002 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-43294-4 |editor-last=Muga |editor-first=J. G. |volume=72 |location=Berlin, Heidelberg |pages=69–98 |language=en |chapter=The Time-Energy Uncertainty Relation |doi=10.1007/3-540-45846-8_3 |editor-last2=Mayato |editor-first2=R. Sala |editor-last3=Egusquiza |editor-first3=I. L.}}</ref> However, one well-known application is both well established<ref>{{Cite book |last=Wigner |first=E. P. |chapter=On the Time–Energy Uncertainty Relation |date=1997 |editor-last=Wightman |editor-first=Arthur S. |title=Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics |chapter-url=http://link.springer.com/10.1007/978-3-662-09203-3_58 |language=en |location=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |pages=538–548 |doi=10.1007/978-3-662-09203-3_58 |isbn=978-3-642-08179-8}}</ref><ref name=Hilgevoord/> and experimentally verified:<ref>{{Cite journal |last1=Lynch |first1=F. J. |last2=Holland |first2=R. E. |last3=Hamermesh |first3=M. |date=1960-10-15 |title=Time Dependence of Resonantly Filtered Gamma Rays from Fe 57 |url=https://link.aps.org/doi/10.1103/PhysRev.120.513 |journal=Physical Review |language=en |volume=120 |issue=2 |pages=513–520 |doi=10.1103/PhysRev.120.513 |issn=0031-899X}}</ref><ref>{{cite book
 | last = Frauenfelder |first=H.
 | year = 1962
 | title = The Mössbauer Effect
 | url = https://archive.org/details/mssbauereffec00frau | publisher = [[W. A. Benjamin]]
 | lccn = 61018181|page=66
}}</ref> the connection between the life-time of a resonance state, <math>\tau_{\sqrt{1/2}}</math> and its energy width <math>\Delta E</math>:
<math display=block>\tau_{\sqrt{1/2}} \Delta E = \pi\hbar/4.</math>
In particle-physics, widths from experimental fits to the [[Relativistic Breit–Wigner distribution|Breit–Wigner energy distribution]] are used to characterize the lifetime of quasi-stable or decaying states.<ref>{{Cite journal |last1=Bohm |first1=Arno R. |last2=Sato |first2=Yoshihiro |date=2005-04-28 |title=Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution |url=https://link.aps.org/doi/10.1103/PhysRevD.71.085018 |journal=Physical Review D |language=en |volume=71 |issue=8 |page=085018 |arxiv=hep-ph/0412106 |doi=10.1103/PhysRevD.71.085018 |bibcode=2005PhRvD..71h5018B |s2cid=119417992 |issn=1550-7998}}</ref>

An informal, heuristic meaning of the principle is the following:<ref>Karplus, Martin, and Porter, Richard Needham (1970). ''Atoms and Molecules''. California: Benjamin Cummings. p. 68 {{ISBN|978-0805352184}}. {{oclc|984466711}}</ref> A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. For example, in [[Electromagnetic spectroscopy|spectroscopy]], excited states have a finite lifetime. By the time–energy uncertainty principle, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the [[Spectral linewidth|''natural linewidth'']]. Fast-decaying states have a broad linewidth, while slow-decaying states have a narrow linewidth.<ref>The broad linewidth of fast-decaying states makes it difficult to accurately measure the energy of the state, and researchers have even used detuned microwave cavities to slow down the decay rate, to get sharper peaks. {{Cite journal |last=Gabrielse |first=Gerald |author2=H. Dehmelt |title=Observation of Inhibited Spontaneous Emission |journal=Physical Review Letters |volume=55 |pages=67–70 |year=1985 |doi=10.1103/PhysRevLett.55.67 |pmid=10031682 |issue=1 |bibcode=1985PhRvL..55...67G}}</ref> The same linewidth effect also makes it difficult to specify the [[rest mass]] of unstable, fast-decaying particles in [[particle physics]]. The faster the [[particle decay]]s (the shorter its lifetime), the less certain is its mass (the larger the particle's [[Resonance (particle physics)|width]]).

===Time in quantum mechanics===
The concept of "time" in quantum mechanics offers many challenges.<ref name=HilgevoordConfusion/> There is no quantum theory of time measurement; relativity is both fundamental to time and difficult to include in quantum mechanics.<ref name="Busch2002"/> While position and momentum are associated with a single particle, time is a system property: it has no operator needed for the Robertson–Schrödinger relation.<ref name=Sen2014/> The mathematical treatment of stable and unstable quantum systems differ.<ref>{{Cite journal |last=Bohm |first=Arno |date=January 2011 |title=Resonances/decaying states and the mathematics of quantum physics |url=https://linkinghub.elsevier.com/retrieve/pii/S0034487711600189 |journal=Reports on Mathematical Physics |language=en |volume=67 |issue=3 |pages=279–303 |doi=10.1016/S0034-4877(11)60018-9 |bibcode=2011RpMP...67..279B |access-date=2024-01-24 |archive-date=2023-12-04 |archive-url=https://web.archive.org/web/20231204062259/https://linkinghub.elsevier.com/retrieve/pii/S0034487711600189 |url-status=live }}</ref> These factors combine to make energy–time uncertainty principles controversial.

Three notions of "time" can be distinguished:<ref name="Busch2002"/> external, intrinsic, and observable. External or laboratory time is seen by the experimenter; intrinsic time is inferred by changes in dynamic variables, like the hands of a clock or the motion of a free particle; observable time concerns time as an observable, the measurement of time-separated events.

An external-time energy–time uncertainty principle might say that measuring the energy of a quantum system to an accuracy <math>\Delta E</math> requires a time interval <math>\Delta t > h/\Delta E</math>.<ref name=Hilgevoord>{{Cite journal |last=Hilgevoord |first=Jan |date=1996-12-01 |title=The uncertainty principle for energy and time |url=https://pubs.aip.org/ajp/article/64/12/1451/1054748/The-uncertainty-principle-for-energy-and-time |journal=American Journal of Physics |language=en |volume=64 |issue=12 |pages=1451–1456 |doi=10.1119/1.18410 |bibcode=1996AmJPh..64.1451H |issn=0002-9505 |access-date=2023-11-12 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223155750/https://pubs.aip.org/aapt/ajp/article-abstract/64/12/1451/1054748/The-uncertainty-principle-for-energy-and-time?redirectedFrom=fulltext |url-status=live }}</ref> However, [[Yakir Aharonov]] and [[David Bohm]]<ref>{{Cite journal |url=http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Aharonov%20a.pdf |title=Time in the Quantum Theory and the Uncertainty Relation for Time and Energy |journal=Physical Review |volume=122 |issue=5 |date=June 1, 1961 |first1=Y. |last1=Aharonov |first2=D. |last2=Bohm |pages=1649–1658 |doi=10.1103/PhysRev.122.1649 |bibcode=1961PhRv..122.1649A |access-date=2012-01-21 |archive-date=2014-01-09 |archive-url=https://web.archive.org/web/20140109081758/http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Aharonov%20a.pdf |url-status=dead }}</ref><ref name="Busch2002"/> have shown that, in some quantum systems, energy can be measured accurately within an arbitrarily short time: external-time uncertainty principles are not universal.

Intrinsic time is the basis for several formulations of energy–time uncertainty relations, including the Mandelstam–Tamm relation discussed in the next section. A physical system with an intrinsic time closely matching the external laboratory time is called a "clock".<ref name=HilgevoordConfusion>{{Cite journal |last=Hilgevoord |first=Jan |date=March 2005 |title=Time in quantum mechanics: a story of confusion |url=https://linkinghub.elsevier.com/retrieve/pii/S1355219804000760 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |language=en |volume=36 |issue=1 |pages=29–60 |doi=10.1016/j.shpsb.2004.10.002 |bibcode=2005SHPMP..36...29H |access-date=2024-01-28 |archive-date=2022-10-23 |archive-url=https://web.archive.org/web/20221023233903/https://linkinghub.elsevier.com/retrieve/pii/S1355219804000760 |url-status=live }}</ref>{{rp|31}}

Observable time, measuring time between two events, remains a challenge for quantum theories; some progress has been made using [[POVM| positive operator-valued measure]] concepts.<ref name="Busch2002"/>

===Mandelstam–Tamm===
In 1945, [[Leonid Mandelstam]] and [[Igor Tamm]] derived a non-relativistic ''time–energy uncertainty relation'' as follows.<ref>L. I. Mandelstam, I. E. Tamm, [http://daarb.narod.ru/mandtamm/index-eng.html ''The uncertainty relation between energy and time in nonrelativistic quantum mechanics''] {{Webarchive|url=https://web.archive.org/web/20190607131054/http://daarb.narod.ru/mandtamm/index-eng.html |date=2019-06-07 }}, 1945.</ref><ref name="Busch2002"/> From Heisenberg mechanics, the generalized [[Ehrenfest theorem]] for an observable ''B'' without explicit time dependence, represented by a self-adjoint operator <math>\hat B</math> relates time dependence of the average value of <math>\hat B</math> to the average of its commutator with the Hamiltonian:

<math display=block> \frac{d\langle \hat{B} \rangle}{dt} = \frac{i}{\hbar}\langle [\hat{H},\hat{B}]\rangle. </math>

The value of <math>\langle [\hat{H},\hat{B}]\rangle</math> is then substituted in the [[#Robertson–Schrödinger_uncertainty_relations|Robertson uncertainty relation]] for the energy operator <math>\hat H</math> and <math>\hat B</math>:
<math display=block> \sigma_H\sigma_B \geq \left|\frac{1}{2i} \langle[ \hat{H}, \hat{B}] \rangle\right|, </math>
giving
<math display="block"> \sigma_H \frac{\sigma_B}{\left| \frac{d\langle \hat B \rangle}{dt}\right |} \ge \frac{\hbar}{2}</math>
(whenever the denominator is nonzero).
While this is a universal result, it depends upon the observable chosen and that the deviations <math>\sigma_H</math> and <math>\sigma_B</math> are computed for a particular state.
Identifying <math>\Delta E \equiv \sigma_E </math> and the characteristic time
<math display="block">\tau_B \equiv \frac{\sigma_B}{\left| \frac{d\langle \hat B \rangle}{dt}\right |}</math>
gives an energy–time relationship <math>\Delta E \tau_B \ge \frac{\hbar}{2}.</math>
Although <math>\tau_B</math> has the dimension of time, it is different from the time parameter ''t'' that enters the [[Schrödinger equation]]. This <math>\tau_B</math> can be interpreted as time for which the expectation value of the observable, <math>\langle \hat B \rangle,</math> changes by an amount equal to one standard deviation.<ref>{{Cite book |last=Naber |first=Gregory L. |url=https://books.google.com/books?id=kARGEAAAQBAJ |title=Quantum Mechanics: An Introduction to the Physical Background and Mathematical Structure |year=2021 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-075194-9 |pages=230 |language=en |access-date=2024-01-20 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223155539/https://books.google.com/books?id=kARGEAAAQBAJ |url-status=live }}</ref>
Examples:
* The time a free quantum particle passes a point in space is more uncertain as the energy of the state is more precisely controlled: <math>\Delta T = \hbar/2\Delta E.</math> Since the time spread is related to the particle position spread and the energy spread is related to the momentum spread, this relation is directly related to position–momentum uncertainty.<ref name="GriffithsSchroeter2018" />{{rp|144}}
* A [[Delta particle]], a quasistable composite of quarks related to protons and neutrons, has a lifetime of 10<sup>−23</sup>&nbsp;s, so its measured [[Mass–energy equivalence| mass equivalent to energy]], 1232&nbsp;MeV/''c''<sup>2</sup>, varies by ±120&nbsp;MeV/''c''<sup>2</sup>; this variation is intrinsic and not caused by measurement errors.<ref name="GriffithsSchroeter2018" />{{rp|144}}
* Two energy states <math>\psi_{1,2}</math> with energies <math>E_{1,2},</math> superimposed to create a composite state
:<math display="block">\Psi(x,t) = a\psi_1(x) e^{-iE_1t/h} + b\psi_2(x) e^{-iE_2t/h}.</math>
:The probability amplitude of this state has a time-dependent interference term:
:<math display="block">|\Psi(x,t)|^2 = a^2|\psi_1(x)|^2 + b^2|\psi_2(x)|^2 + 2ab\cos(\frac{E_2 - E_1}{\hbar}t).</math>
:The oscillation period varies inversely with the energy difference: <math>\tau = 2\pi\hbar/(E_2 - E_1)</math>.<ref name="GriffithsSchroeter2018" />{{rp|144}}
Each example has a different meaning for the time uncertainty, according to the observable and state used.

===Quantum field theory===

Some formulations of [[quantum field theory]] uses temporary electron–positron pairs in its calculations called [[virtual particles]]. The mass-energy and lifetime of these particles are related by the energy–time uncertainty relation. The energy of a quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of ''all histories'' must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.

The energy–time uncertainty principle does not temporarily violate [[conservation of energy]]; it does not imply that energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time.<ref name="GriffithsSchroeter2018" />{{rp|145}} The energy of the universe is not an exactly known parameter at all times.<ref name=Sen2014/> When events transpire at very short time intervals, there is uncertainty in the energy of these events.

==Harmonic analysis==
{{Main article|Fourier transform#Uncertainty principle}}
In the context of [[harmonic analysis]] the uncertainty principle implies that one cannot at the same time localize the value of a function and its Fourier transform. To wit, the following inequality holds,
<math display="block">\left(\int_{-\infty}^\infty x^2 |f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty \xi^2 |\hat{f}(\xi)|^2\,d\xi\right)\ge \frac{\|f\|_2^4}{16\pi^2}.</math>

Further mathematical uncertainty inequalities, including the above [[entropic uncertainty]], hold between a function {{mvar|f}} and its Fourier transform {{math| ƒ̂}}:<ref>{{Citation|first1=V.|last1=Havin|first2= B.|last2=Jöricke|title=The Uncertainty Principle in Harmonic Analysis|publisher=Springer-Verlag|year=1994}}</ref><ref>{{Citation | last1 = Folland | first1 = Gerald | last2 = Sitaram |first2 = Alladi | title = The Uncertainty Principle: A Mathematical Survey | journal = Journal of Fourier Analysis and Applications | date = May 1997 | volume = 3 | issue = 3 | pages = 207–238 | doi = 10.1007/BF02649110 | bibcode = 1997JFAA....3..207F | mr=1448337 | s2cid = 121355943 }}</ref><ref>{{springer|title=Uncertainty principle, mathematical|id=U/u130020|first=A|last=Sitaram|year=2001}}</ref>
<math display="block">H_x+H_\xi \ge \log(e/2)</math>

===Signal processing {{anchor|Gabor limit}}===
In the context of [[time–frequency analysis]] uncertainty principles are referred to as the '''Gabor limit''', after [[Dennis Gabor]], or sometimes the ''Heisenberg–Gabor limit''. The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be both [[time limited]] and [[band limited]] (a function and its Fourier transform cannot both have bounded domain)—see [[Bandlimiting#Bandlimited versus timelimited|bandlimited versus timelimited]]. More accurately, the ''time-bandwidth'' or ''duration-bandwidth'' product satisfies
<math display="block">\sigma_{t} \sigma_{f} \ge \frac{1}{4\pi} \approx 0.08 \text{ cycles},</math>
where <math>\sigma_{t}</math> and <math>\sigma_{f}</math> are the standard deviations of the time and frequency energy concentrations respectively.<ref>{{cite book | last=Mallat | first=S. G. | title=A wavelet tour of signal processing: the sparse way | publisher=Elsevier/Academic Press | publication-place=Amsterdam ; Boston | date=2009 | isbn=978-0-12-374370-1|doi=10.1016/B978-0-12-374370-1.X0001-8|page=44}}</ref> The minimum is attained for a [[Gaussian function|Gaussian]]-shaped pulse ([[Gabor wavelet]]) [For the un-squared Gaussian (i.e. signal amplitude) and its un-squared Fourier transform magnitude <math>\sigma_t\sigma_f=1/2\pi</math>; squaring reduces each <math>\sigma</math> by a factor <math>\sqrt 2</math>.] Another common measure is the product of the time and frequency [[full width at half maximum]] (of the power/energy), which for the Gaussian equals <math>2 \ln 2 / \pi \approx 0.44</math> (see [[bandwidth-limited pulse]]).

Stated differently, one cannot simultaneously sharply localize a signal {{mvar|f}} in both the [[time domain]] and [[frequency domain]].

When applied to [[Filter (signal processing)|filters]], the result implies that one cannot simultaneously achieve a high temporal resolution and high frequency resolution at the same time; a concrete example are the [[Short-time Fourier transform#Resolution issues|resolution issues of the short-time Fourier transform]]—if one uses a wide window, one achieves good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite trade-off.

Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) time–frequency plane. In practice, the Gabor limit limits the ''simultaneous'' time–frequency resolution one can achieve without interference; it is possible to achieve higher resolution, but at the cost of different components of the signal interfering with each other.

As a result, in order to analyze signals where the [[Transient (acoustics)|transients]] are important, the [[Wavelet Transform|wavelet transform]] is often used instead of the Fourier.

===Discrete Fourier transform===
Let <math>\left \{ \mathbf{ x_n } \right \} := x_0, x_1, \ldots, x_{N-1}</math> be a sequence of ''N'' complex numbers and <math>\left \{ \mathbf{X_k} \right \} := X_0, X_1, \ldots, X_{N-1},</math> be its [[Discrete Fourier transform#Uncertainty principles | discrete Fourier transform]].

Denote by <math>\|x\|_0</math> the number of non-zero elements in the time sequence <math>x_0,x_1,\ldots,x_{N-1}</math> and by <math>\|X\|_0</math> the number of non-zero elements in the frequency sequence <math>X_0,X_1,\ldots,X_{N-1}</math>. Then,
<math display="block">\|x\|_0 \cdot \|X\|_0 \ge N.</math>

This inequality is [[inequality (mathematics)#Sharp inequalities|sharp]], with equality achieved when ''x'' or ''X'' is a Dirac mass, or more generally when ''x'' is a nonzero multiple of a Dirac comb supported on a subgroup of the integers modulo ''N'' (in which case ''X'' is also a Dirac comb supported on a complementary subgroup, and vice versa).

More generally, if ''T'' and ''W'' are subsets of the integers modulo ''N'', let <math>L_T,R_W : \ell^2(\mathbb Z/N\mathbb Z)\to\ell^2(\mathbb Z/N\mathbb Z)</math> denote the time-limiting operator and [[bandlimiting|band-limiting operator]]s, respectively. Then
<math display="block">\|L_TR_W\|^2 \le \frac{|T||W|}{|G|} </math>
where the norm is the [[operator norm]] of operators on the Hilbert space <math>\ell^2(\mathbb Z/N\mathbb Z)</math> of functions on the integers modulo ''N''. This inequality has implications for [[signal reconstruction]].<ref name="Donoho">{{cite journal |last1=Donoho |first1=D.L. |last2=Stark |first2=P.B |year=1989 |title=Uncertainty principles and signal recovery |journal=SIAM Journal on Applied Mathematics |volume=49 |issue=3 |pages=906–931 |doi=10.1137/0149053}}</ref>

When ''N'' is a [[prime number]], a stronger inequality holds:
<math display="block">\|x\|_0 + \|X\|_0 \ge N + 1.</math>
Discovered by [[Terence Tao]], this inequality is also sharp.<ref>{{citation|
journal=Mathematical Research Letters
|volume=12
|year=2005
|issue=1
|title=An uncertainty principle for cyclic groups of prime order
|pages=121–127
|author=[[Terence Tao]]
|doi=10.4310/MRL.2005.v12.n1.a11
|arxiv=math/0308286
|s2cid=8548232
}}</ref>

=== Benedicks's theorem ===
Amrein–Berthier<ref>
{{citation
 | last1 = Amrein | first1 = W.O.
 | last2 = Berthier | first2 = A.M.
 | year = 1977
 | title = On support properties of ''L''<sup>''p''</sup>-functions and their Fourier transforms
 | journal = Journal of Functional Analysis
 | volume = 24 | issue = 3 | pages = 258–267
 | doi = 10.1016/0022-1236(77)90056-8
 | postscript = .
 | doi-access = free
}}</ref> and Benedicks's theorem<ref>{{citation |first=M. |last=Benedicks |author-link=Michael Benedicks |title=On Fourier transforms of functions supported on sets of finite Lebesgue measure |journal=J. Math. Anal. Appl. |volume=106 |year=1985 |issue=1 |pages=180–183 |doi=10.1016/0022-247X(85)90140-4 |doi-access=free }}</ref> intuitively says that the set of points where {{mvar|f}} is non-zero and the set of points where {{math|ƒ̂}} is non-zero cannot both be small.

Specifically, it is impossible for a function {{mvar|f}} in {{math|''L''<sup>2</sup>('''R''')}} and its Fourier transform {{math|ƒ̂}} to both be [[support of a function|supported]] on sets of finite [[Lebesgue measure]]. A more quantitative version is<ref>{{Citation|first=F.|last=Nazarov|author-link=Fedor Nazarov|title=Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type|journal=St. Petersburg Math. J.|volume=5|year=1994|pages=663–717}}</ref><ref>{{Citation|first=Ph.|last=Jaming|title=Nazarov's uncertainty principles in higher dimension|journal= J. Approx. Theory|volume=149|year=2007|issue=1|pages=30–41|doi=10.1016/j.jat.2007.04.005|arxiv=math/0612367|s2cid=9794547}}</ref>
<math display="block">\|f\|_{L^2(\mathbf{R}^d)}\leq Ce^{C|S||\Sigma|} \bigl(\|f\|_{L^2(S^c)} + \| \hat{f} \|_{L^2(\Sigma^c)} \bigr) ~.</math>

One expects that the factor {{math|''Ce''<sup>''C''{{abs|''S''}}{{abs|''Σ''}}</sup>}} may be replaced by {{math|''Ce''<sup>''C''({{abs|''S''}}{{abs|''Σ''}})<sup>1/''d''</sup></sup>}}, which is only known if either {{mvar|S}} or {{mvar|Σ}} is convex.

=== Hardy's uncertainty principle ===
The mathematician [[G. H. Hardy]] formulated the following uncertainty principle:<ref>{{Citation|first=G.H.|last=Hardy|author-link=G. H. Hardy|title=A theorem concerning Fourier transforms|journal=Journal of the London Mathematical Society|volume=8|year=1933|issue=3|pages=227–231|doi=10.1112/jlms/s1-8.3.227}}</ref> it is not possible for {{mvar|f}} and {{math| ƒ̂}} to both be "very rapidly decreasing". Specifically, if {{mvar|f}} in <math>L^2(\mathbb{R})</math> is such that
<math display="block">|f(x)|\leq C(1+|x|)^Ne^{-a\pi x^2}</math>
and
<math display="block">|\hat{f}(\xi)|\leq C(1+|\xi|)^Ne^{-b\pi \xi^2}</math> (<math>C>0,N</math> an integer),
then, if {{math|1=''ab'' > 1, ''f'' = 0}}, while if {{math|1=''ab'' = 1}}, then there is a polynomial {{mvar|P}} of degree {{math|≤ ''N''}} such that
<math display="block">f(x)=P(x)e^{-a\pi x^2}. </math>

This was later improved as follows: if <math>f \in L^2(\mathbb{R}^d)</math> is such that
<math display="block">\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}|f(x)||\hat{f}(\xi)|\frac{e^{\pi|\langle x,\xi\rangle|}}{(1+|x|+|\xi|)^N} \, dx \, d\xi < +\infty ~,</math>
then
<math display="block">f(x)=P(x)e^{-\pi\langle Ax,x\rangle} ~,</math>
where {{mvar|P}} is a polynomial of degree {{math|(''N'' − ''d'')/2}} and {{mvar|A}} is a real {{math|''d'' × ''d''}} positive definite matrix.

This result was stated in Beurling's complete works without proof and proved in Hörmander<ref>{{Citation | first=L. | last=Hörmander | author-link=Lars Hörmander|title=A uniqueness theorem of Beurling for Fourier transform pairs|journal= Ark. Mat. | volume=29|issue=1–2|year=1991|pages=231–240|bibcode=1991ArM....29..237H|doi=10.1007/BF02384339|s2cid=121375111 | doi-access=free}}</ref> (the case <math>d=1,N=0</math>) and Bonami, Demange, and Jaming<ref>{{Citation | first1=A. | last1=Bonami | author1-link= Aline Bonami |first2=B.|last2=Demange|first3=Ph.|last3=Jaming|title=Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms |journal= Rev. Mat. Iberoamericana | volume=19 | year=2003 | pages=23–55 | bibcode=2001math......2111B|arxiv=math/0102111| doi=10.4171/RMI/337|s2cid=1211391}}</ref> for the general case. Note that Hörmander–Beurling's version implies the case {{math|''ab'' > 1}} in Hardy's Theorem while the version by Bonami–Demange–Jaming covers the full strength of Hardy's Theorem. A different proof of Beurling's theorem based on Liouville's theorem appeared in ref.<ref>{{Citation|first=Haakan|last=Hedenmalm|title=Heisenberg's uncertainty principle in the sense of Beurling|journal=[[Journal d'Analyse Mathématique]] | volume=118 | issue=2 | year=2012 | pages=691–702 | doi=10.1007/s11854-012-0048-9 | doi-access=free | arxiv=1203.5222 | bibcode=2012arXiv1203.5222H | s2cid=54533890}}</ref>

A full description of the case {{math|''ab'' < 1}} as well as the following extension to Schwartz class distributions appears in ref.<ref>{{Citation|first=Bruno|last=Demange|title=Uncertainty Principles Associated to Non-degenerate Quadratic Forms|year=2009|publisher= Société Mathématique de France|isbn=978-2-85629-297-6}}</ref>

{{math theorem| If a tempered distribution <math>f\in\mathcal{S}'(\R^d)</math> is such that
<math display="block">e^{\pi|x|^2}f\in\mathcal{S} '(\R^d)</math>
and
<math display="block">e^{\pi|\xi|^2}\hat f\in\mathcal{S}'(\R^d) ~,</math>
then
<math display="block">f(x)=P(x)e^{-\pi\langle Ax,x\rangle} ~,</math>
for some convenient polynomial {{mvar|P}} and real positive definite matrix {{mvar|A}} of type {{math|''d'' × ''d''}}.}}

==Additional uncertainty relations==

===Heisenberg limit===
In [[quantum metrology]], and especially [[interferometry]], the '''Heisenberg limit''' is the optimal rate at which the accuracy of a measurement can scale with the energy used in the measurement. Typically, this is the measurement of a phase (applied to one arm of a [[beam-splitter]]) and the energy is given by the number of photons used in an [[interferometer]]. Although some claim to have broken the Heisenberg limit, this reflects disagreement on the definition of the scaling resource.<ref>{{Cite journal | last1 = Giovannetti | first1 = V. | last2 = Lloyd | first2 = S. | last3 = Maccone | first3 = L. | doi = 10.1038/nphoton.2011.35 | title = Advances in quantum metrology | journal = Nature Photonics | volume = 5 | issue = 4 | pages = 222 | year = 2011 | arxiv = 1102.2318 | bibcode = 2011NaPho...5..222G | s2cid = 12591819 }}; [https://arxiv.org/abs/1102.2318 arXiv] {{Webarchive|url=https://web.archive.org/web/20200806200530/https://arxiv.org/abs/1102.2318 |date=2020-08-06 }}</ref> Suitably defined, the Heisenberg limit is a consequence of the basic principles of quantum mechanics and cannot be beaten, although the weak Heisenberg limit can be beaten.<ref>{{Cite journal|last=Luis|first=Alfredo|date=2017-03-13|title=Breaking the weak Heisenberg limit | journal=Physical Review A | language=en|volume=95|issue=3 | pages=032113 | doi=10.1103/PhysRevA.95.032113 | arxiv=1607.07668 | bibcode=2017PhRvA..95c2113L | s2cid=55838380|issn=2469-9926}}</ref>

===Systematic and statistical errors===

The inequalities above focus on the ''statistical imprecision'' of observables as quantified by the standard deviation <math>\sigma</math>. Heisenberg's original version, however, was dealing with the ''systematic error'', a disturbance of the quantum system produced by the measuring apparatus, i.e., an observer effect.

If we let <math>\varepsilon_A</math> represent the error (i.e., [[accuracy|inaccuracy]]) of a measurement of an observable ''A'' and <math>\eta_B</math> the disturbance produced on a subsequent measurement of the conjugate variable ''B'' by the former measurement of ''A'', then the inequality proposed by Masanao Ozawa − encompassing both systematic and statistical errors - holds:<ref name="Ozawa2003"/>
{{Equation box 1
|indent =:
|equation = <math> \varepsilon_A\, \eta_B + \varepsilon_A \, \sigma_B + \sigma_A \, \eta_B \,\ge\, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

Heisenberg's uncertainty principle, as originally described in the 1927 formulation, mentions only the first term of Ozawa inequality, regarding the ''systematic error''. Using the notation above to describe the ''error/disturbance'' effect of ''sequential measurements'' (first ''A'', then ''B''), it could be written as
{{Equation box 1
|indent =:
|equation = <math> \varepsilon_{A} \, \eta_{B} \, \ge \, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
The formal derivation of the Heisenberg relation is possible but far from intuitive. It was ''not'' proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years.<ref>{{Cite journal | doi = 10.1103/PhysRevLett.111.160405| title = Proof of Heisenberg's Error-Disturbance Relation| journal = Physical Review Letters| volume = 111| issue = 16| year = 2013| last1 = Busch | first1 = P. | last2 = Lahti | first2 = P. | last3 = Werner | first3 = R. F. |arxiv = 1306.1565 |bibcode = 2013PhRvL.111p0405B | pmid=24182239 | page=160405| s2cid = 24507489}}</ref><ref>{{Cite journal | doi = 10.1103/PhysRevA.89.012129| title = Heisenberg uncertainty for qubit measurements| journal = Physical Review A| volume = 89| issue = 1| pages = 012129| year = 2014| last1 = Busch | first1 = P. | last2 = Lahti | first2 = P. | last3 = Werner | first3 = R. F. |arxiv = 1311.0837 |bibcode = 2014PhRvA..89a2129B | s2cid = 118383022}}</ref>
Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors <math>\sigma_A</math> and <math>\sigma_B</math>. There is increasing experimental evidence<ref name="Rozema"/><ref>{{Cite journal| last1 = Erhart | first1 = J.| last2 = Sponar | first2 =S.| last3 = Sulyok | first3 = G. | last4 = Badurek | first4 = G. | last5 = Ozawa | first5 = M. | last6 = Hasegawa | first6 = Y.| title = Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements | journal = Nature Physics | volume=8 | pages=185–189 | year=2012 | doi=10.1038/nphys2194 | arxiv = 1201.1833 | bibcode = 2012NatPh...8..185E | issue=3 | s2cid = 117270618}}</ref><ref>{{Cite journal| last1 = Baek | first1 = S.-Y. | last2 = Kaneda | first2 = F. | last3 = Ozawa | first3 = M. | last4 = Edamatsu | first4 = K. | title = Experimental violation and reformulation of the Heisenberg's error-disturbance uncertainty relation |journal = Scientific Reports |volume= 3 |pages= 2221 |year= 2013 |doi= 10.1038/srep02221 |bibcode = 2013NatSR...3.2221B | pmid=23860715 | pmc=3713528}}</ref><ref>{{Cite journal| last1 = Ringbauer | first1 = M. | last2 = Biggerstaff | first2 = D.N. | last3 = Broome | first3 = M.A. | last4 = Fedrizzi | first4 = A. | last5 = Branciard | first5 = C. | last6 = White | first6 = A.G. | title = Experimental Joint Quantum Measurements with Minimum Uncertainty |journal = Physical Review Letters |volume= 112 | issue = 2 |pages= 020401 |year= 2014 |doi= 10.1103/PhysRevLett.112.020401 |arxiv = 1308.5688 |bibcode = 2014PhRvL.112b0401R | pmid=24483993| s2cid = 18730255 }}</ref> that the total quantum uncertainty cannot be described by the Heisenberg term alone, but requires the presence of all the three terms of the Ozawa inequality.

Using the same formalism,<ref name="Sen2014"/> it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case of ''simultaneous measurements'' (''A'' and ''B'' at the same time):
{{Equation box 1
|indent =:
|equation = <math> \varepsilon_A \, \varepsilon_B \, \ge \, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
The two simultaneous measurements on ''A'' and ''B'' are necessarily<ref>{{Cite journal | last1 = Björk | first1 = G. | last2 = Söderholm | first2 = J. | last3 = Trifonov | first3 = A. | last4 = Tsegaye | first4 = T. | last5 = Karlsson | first5 = A. |
title = Complementarity and the uncertainty relations | doi = 10.1103/PhysRevA.60.1874 | journal = Physical Review | volume = A60 | issue = 3 | year = 1999| page = 1878 |arxiv = quant-ph/9904069 |bibcode = 1999PhRvA..60.1874B | s2cid = 27371899 }}</ref> ''unsharp'' or [[weak measurement|''weak'']].

It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson<ref name="Sen2014"/>
{{Equation box 1
|indent =:
|equation = <math> \sigma_{A} \, \sigma_{B} \, \ge \, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B} \bigr] \Bigr\rangle \right|</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
and Ozawa relations we obtain
<math display="block">\varepsilon_A \eta_B + \varepsilon_A \, \sigma_B + \sigma_A \, \eta_B + \sigma_A \sigma_B \geq \left|\Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right| .</math>
The four terms can be written as:
<math display="block">(\varepsilon_A + \sigma_A) \, (\eta_B + \sigma_B) \, \geq \, \left|\Bigl\langle\bigl[\hat{A},\hat{B} \bigr] \Bigr\rangle \right| .</math>
Defining:
<math display="block">\bar \varepsilon_A \, \equiv \, (\varepsilon_A + \sigma_A)</math>
as the ''inaccuracy'' in the measured values of the variable ''A'' and
<math display="block">\bar \eta_B \, \equiv \, (\eta_B + \sigma_B)</math>
as the ''resulting fluctuation'' in the conjugate variable ''B'', Kazuo Fujikawa<ref>{{Cite journal|last = Fujikawa|first = Kazuo|title = Universally valid Heisenberg uncertainty relation|journal = Physical Review A|volume=85|year=2012|doi=10.1103/PhysRevA.85.062117|arxiv = 1205.1360 |bibcode = 2012PhRvA..85f2117F|issue=6 |pages = 062117|s2cid = 119640759}}</ref> established an uncertainty relation similar to the Heisenberg original one, but valid both for ''systematic and statistical errors'':
{{Equation box 1
|indent =:
|equation = <math> \bar \varepsilon_A \, \bar \eta_B \, \ge \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

===Quantum entropic uncertainty principle===
For many distributions, the standard deviation is not a particularly natural way of quantifying the structure. For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.<ref name="CarruthersNieto" /><ref>{{Citation |first=D. |last=Judge |title=On the uncertainty relation for angle variables | journal=Il Nuovo Cimento |year=1964|volume=31|issue=2|pages=332–340|doi=10.1007/BF02733639 | bibcode=1964NCim...31..332J | s2cid=120553526 }}</ref><ref>{{Citation |first1= M. |last1= Bouten |first2= N. |last2= Maene | first3= P. | last3= Van Leuven | title=On an uncertainty relation for angle variables | journal=Il Nuovo Cimento | year=1965 | volume=37 | issue=3 | pages=1119–1125 | doi=10.1007/BF02773197 | bibcode=1965NCim...37.1119B | s2cid= 122838645 }}</ref><ref>{{Citation |first=W. H. | last=Louisell | title=Amplitude and phase uncertainty relations|journal=Physics Letters | year=1963 | volume=7 | issue=1 | pages=60–61 | doi=10.1016/0031-9163(63)90442-6 | bibcode = 1963PhL.....7...60L }}</ref> Other examples include highly [[bimodal distribution]]s, or [[unimodal distribution]]s with divergent variance.

A solution that overcomes these issues is an uncertainty based on [[entropic uncertainty]] instead of the product of variances. While formulating the [[many-worlds interpretation]] of quantum mechanics in 1957, [[Hugh Everett III]] conjectured a stronger extension of the uncertainty principle based on entropic certainty.<ref>{{Citation |last1=DeWitt |first1=B. S. |last2=Graham |first2=N. |year=1973 |title=The Many-Worlds Interpretation of Quantum Mechanics |location=Princeton |publisher=[[Princeton University Press]] |pages=52–53 |isbn=0-691-08126-3 }}</ref> This conjecture, also studied by I. I. Hirschman<ref>{{Citation | first=I. I. Jr. |last=Hirschman |title=A note on entropy |journal=[[American Journal of Mathematics]] |year=1957 |volume=79 |issue=1 |pages=152–156 |doi=10.2307/2372390 |postscript=. |jstor=2372390 }}</ref> and proven in 1975 by W. Beckner<ref name="Beckner">{{Citation |first=W. |last=Beckner |title=Inequalities in Fourier analysis |journal=[[Annals of Mathematics]] |volume=102 |issue=6 |year=1975 |pages=159–182 |doi=10.2307/1970980 |postscript=. |jstor=1970980 |pmid=16592223 |pmc=432369 }}</ref> and by Iwo Bialynicki-Birula and Jerzy Mycielski<ref name="BBM">{{Citation |first1=I. |last1=Bialynicki-Birula |last2=Mycielski |first2=J. |title=Uncertainty Relations for Information Entropy in Wave Mechanics |journal=[[Communications in Mathematical Physics]] |volume=44 |year=1975 |pages=129–132 |doi=10.1007/BF01608825 |issue=2 |bibcode=1975CMaPh..44..129B |s2cid=122277352 |url=http://projecteuclid.org/euclid.cmp/1103899297 |access-date=2021-08-17 |archive-date=2021-02-08 |archive-url=https://web.archive.org/web/20210208011223/https://projecteuclid.org/euclid.cmp/1103899297 |url-status=live }}</ref> is that, for two normalized, dimensionless Fourier transform pairs {{math|''f''(''a'')}} and {{math|''g''(''b'')}} where
:<math>f(a) = \int_{-\infty}^\infty g(b)\ e^{2\pi i a b}\,db</math>{{spaces|3}} and {{spaces|3}} <math> \,\,\,g(b) = \int_{-\infty}^\infty f(a)\ e^{- 2\pi i a b}\,da</math>
the Shannon [[Information entropy|information entropies]]
<math display="block">H_a = -\int_{-\infty}^\infty |f(a)|^2 \log |f(a)|^2\,da,</math>
and
<math display="block">H_b = -\int_{-\infty}^\infty |g(b)|^2 \log |g(b)|^2\,db</math>
are subject to the following constraint,
{{Equation box 1
|indent =:
|equation =<math>H_a + H_b \ge \log (e/2)</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where the logarithms may be in any base.

The probability distribution functions associated with the position wave function {{math|''ψ''(''x'')}} and the momentum wave function {{math|''φ''(''x'')}} have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless by
<math display="block">H_x = - \int |\psi(x)|^2 \ln \left(x_0 \, |\psi(x)|^2 \right) dx =-\left\langle \ln \left(x_0 \, \left|\psi(x)\right|^2 \right) \right\rangle</math>
<math display="block">H_p = - \int |\varphi(p)|^2 \ln (p_0\,|\varphi(p)|^2) \,dp =-\left\langle \ln (p_0\left|\varphi(p)\right|^2 ) \right\rangle</math>
where {{math|''x''<sub>0</sub>}} and {{math|''p''<sub>0</sub>}} are some arbitrarily chosen length and momentum respectively, which render the arguments of the logarithms dimensionless. Note that the entropies will be functions of these chosen parameters. Due to the [[Wavefunction#Relation between wave functions|Fourier transform relation]] between the position wave function {{math|''ψ''(''x'')}} and the momentum wavefunction {{math|''φ''(''p'')}}, the above constraint can be written for the corresponding entropies as
{{Equation box 1
|indent =:
|equation = <math>H_x + H_p \ge \log \left(\frac{e\,h}{2\,x_0\,p_0}\right)</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where {{mvar|h}} is the [[Planck constant]].

Depending on one's choice of the {{math|''x<sub>0</sub> p<sub>0</sub>''}} product, the expression may be written in many ways. If {{math|''x''<sub>0</sub> ''p''<sub>0</sub>}} is chosen to be {{mvar|h}}, then
<math display="block">H_x + H_p \ge \log \left(\frac{e}{2}\right)</math>

If, instead, {{math|''x''<sub>0</sub> ''p''<sub>0</sub>}} is chosen to be {{mvar|ħ}}, then
<math display="block">H_x + H_p \ge \log (e\,\pi)</math>

If {{math|''x''<sub>0</sub>}} and {{math|''p''<sub>0</sub>}} are chosen to be unity in whatever system of units are being used, then
<math display="block">H_x + H_p \ge \log \left(\frac{e\,h }{2}\right)</math>
where {{mvar|h}} is interpreted as a dimensionless number equal to the value of the Planck constant in the chosen system of units. Note that these inequalities can be extended to multimode quantum states, or wavefunctions in more than one spatial dimension.<ref>{{cite journal |last1=Huang |first1=Yichen |title=Entropic uncertainty relations in multidimensional position and momentum spaces | journal=Physical Review A |date=24 May 2011 |volume=83 |issue=5 |page=052124 | doi=10.1103/PhysRevA.83.052124 | bibcode=2011PhRvA..83e2124H | arxiv=1101.2944 | s2cid=119243096 }}</ref>

The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse [[logarithmic Sobolev inequalities]]<ref>{{citation |first=D. |last=Chafaï |chapter=Gaussian maximum of entropy and reversed log-Sobolev inequality|arxiv=math/0102227 |doi=10.1007/978-3-540-36107-7_5 |year=2003 |isbn=978-3-540-00072-3 |pages=194–200|title=Séminaire de Probabilités XXXVI |volume=1801 |series=Lecture Notes in Mathematics |s2cid=17795603 }}</ref>
<math display="block">H_x \le \frac{1}{2} \log ( 2e\pi \sigma_x^2 / x_0^2 )~,</math>
<math display="block">H_p \le \frac{1}{2} \log ( 2e\pi \sigma_p^2 /p_0^2 )~,</math>
(equivalently, from the fact that normal distributions maximize the entropy of all such with a given variance), it readily follows that this entropic uncertainty principle is ''stronger than the one based on standard deviations'', because
<math display="block">\sigma_x \sigma_p \ge \frac{\hbar}{2} \exp\left(H_x + H_p - \log \left(\frac{e\,h}{2\,x_0\,p_0}\right)\right) \ge \frac{\hbar}{2}~.</math>

In other words, the Heisenberg uncertainty principle, is a consequence of the quantum entropic uncertainty principle, but not vice versa. A few remarks on these inequalities. First, the choice of [[base e]] is a matter of popular convention in physics. The logarithm can alternatively be in any base, provided that it be consistent on both sides of the inequality. Second, recall the [[Shannon entropy]] has been used, ''not'' the quantum [[von Neumann entropy]]. Finally, the normal distribution saturates the inequality, and it is the only distribution with this property, because it is the [[maximum entropy probability distribution]] among those with fixed variance (cf. [[differential entropy#Maximization in the normal distribution|here]] for proof).

{| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
!Entropic uncertainty of the normal distribution
|-
|We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. The length scale can be set to whatever is convenient, so we assign
<math display="block">x_0 = \sqrt{\frac{\hbar}{2m\omega}}</math>
<math display="block">\begin{align}
\psi(x) &= \left(\frac{m \omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \omega x^2}{2\hbar}\right)} \\
&= \left(\frac{1}{2\pi x_0^2}\right)^{1/4} \exp{\left( -\frac{x^2}{4x_0^2}\right)}
\end{align}</math>

The probability distribution is the normal distribution
<math display="block">|\psi(x)|^2 = \frac{1}{x_0 \sqrt{2\pi}} \exp{\left( -\frac{x^2}{2x_0^2}\right)}</math>
with Shannon entropy
<math display="block">\begin{align}
H_x &= - \int |\psi(x)|^2 \ln (|\psi(x)|^2 \cdot x_0 ) \,dx \\
&= -\frac{1}{x_0 \sqrt{2\pi}} \int_{-\infty}^\infty \exp{\left( -\frac{x^2}{2x_0^2}\right)} \ln \left[\frac{1}{\sqrt{2\pi}} \exp{\left( -\frac{x^2}{2x_0^2}\right)}\right] \, dx \\
&= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \exp{\left( -\frac{u^2}{2}\right)} \left[\ln(\sqrt{2\pi}) + \frac{u^2}{2}\right] \, du\\
&= \ln(\sqrt{2\pi}) + \frac{1}{2}.
\end{align}</math>

A completely analogous calculation proceeds for the momentum distribution. Choosing a standard momentum of <math>p_0=\hbar/x_0</math>:
<math display="block">\varphi(p) = \left(\frac{2 x_0^2}{\pi \hbar^2}\right)^{1/4} \exp{\left( -\frac{x_0^2 p^2}{\hbar^2}\right)}</math>
<math display="block">|\varphi(p)|^2 = \sqrt{\frac{2 x_0^2}{\pi \hbar^2}} \exp{\left( -\frac{2x_0^2 p^2}{\hbar^2}\right)}</math>
<math display="block">\begin{align}
H_p &= - \int |\varphi(p)|^2 \ln (|\varphi(p)|^2 \cdot \hbar / x_0 ) \,dp \\
&= -\sqrt{\frac{2 x_0^2}{\pi \hbar^2}} \int_{-\infty}^\infty \exp{\left( -\frac{2x_0^2 p^2}{\hbar^2}\right)} \ln \left[\sqrt{\frac{2}{\pi}} \exp{\left( -\frac{2x_0^2 p^2}{\hbar^2}\right)}\right] \, dp \\
&= \sqrt{\frac{2}{\pi}} \int_{-\infty}^\infty \exp{\left( -2v^2\right)} \left[\ln\left(\sqrt{\frac{\pi}{2}}\right) + 2v^2 \right] \, dv \\
&= \ln\left(\sqrt{\frac{\pi}{2}}\right) + \frac{1}{2}.
\end{align}</math>

The entropic uncertainty is therefore the limiting value
<math display="block">\begin{align}
H_x+H_p &= \ln(\sqrt{2\pi}) + \frac{1}{2} + \ln\left(\sqrt{\frac{\pi}{2}}\right) + \frac{1}{2}\\
&= 1 + \ln \pi = \ln(e\pi).
\end{align}</math>
|}

A measurement apparatus will have a finite resolution set by the discretization of its possible outputs into bins, with the probability of lying within one of the bins given by the Born rule. We will consider the most common experimental situation, in which the bins are of uniform size. Let ''δx'' be a measure of the spatial resolution. We take the zeroth bin to be centered near the origin, with possibly some small constant offset ''c''. The probability of lying within the jth interval of width ''δx'' is
<math display="block">\operatorname P[x_j]= \int_{(j-1/2)\delta x-c}^{(j+1/2)\delta x-c}| \psi(x)|^2 \, dx</math>

To account for this discretization, we can define the Shannon entropy of the wave function for a given measurement apparatus as
<math display="block">H_x=-\sum_{j=-\infty}^\infty \operatorname P[x_j] \ln \operatorname P[x_j].</math>

Under the above definition, the entropic uncertainty relation is
<math display="block">H_x + H_p > \ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x \delta p}{h} \right).</math>

Here we note that {{math|''δx'' ''δp''/''h''}} is a typical infinitesimal phase space volume used in the calculation of a [[partition function (statistical mechanics)|partition function]]. The inequality is also strict and not saturated. Efforts to improve this bound are an active area of research.

{| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
!Normal distribution example
|-
|We demonstrate this method first on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations.

<math display="block">\psi(x)=\left(\frac{m \omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \omega x^2}{2\hbar}\right)}</math>

The probability of lying within one of these bins can be expressed in terms of the [[error function]].

<math display="block">\begin{align}
\operatorname P[x_j] &= \sqrt{\frac{m \omega}{\pi \hbar}} \int_{(j-1/2)\delta x}^{(j+1/2)\delta x} \exp\left( -\frac{m \omega x^2}{\hbar}\right) \, dx \\
&= \sqrt{\frac{1}{\pi}} \int_{(j-1/2)\delta x\sqrt{m \omega / \hbar}}^{(j+1/2)\delta x\sqrt{m \omega / \hbar}} e^{u^2} \, du \\
&= \frac{1}{2} \left[ \operatorname{erf} \left( \left(j+\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right) \right]
\end{align}</math>

The momentum probabilities are completely analogous.

<math display="block">\operatorname P[p_j] = \frac{1}{2} \left[ \operatorname{erf} \left( \left(j+\frac{1}{2}\right)\delta p \cdot \frac{1}{\sqrt{\hbar m \omega}}\right)- \operatorname{erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \frac{1}{\sqrt{\hbar m \omega}}\right) \right]</math>

For simplicity, we will set the resolutions to
<math display="block">\delta x = \sqrt{\frac{h}{m \omega}}</math>
<math display="block">\delta p = \sqrt{h m \omega}</math>
so that the probabilities reduce to
<math display="block">\operatorname P[x_j] = \operatorname P[p_j] = \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right]</math>

The Shannon entropy can be evaluated numerically.
<math display="block">\begin{align}
H_x = H_p &= -\sum_{j=-\infty}^\infty \operatorname P[x_j] \ln \operatorname P[x_j] \\
&= -\sum_{j=-\infty}^\infty \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] \ln \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] \\
&\approx 0.3226
\end{align}</math>

The entropic uncertainty is indeed larger than the limiting value.
<math display="block">H_x + H_p \approx 0.3226 + 0.3226 = 0.6452 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.3069</math>

Note that despite being in the optimal case, the inequality is not saturated.
|}

{| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
!Sinc function example
|-
|An example of a unimodal distribution with infinite variance is the [[sinc function]]. If the wave function is the correctly normalized uniform distribution,
<math display="block">\psi(x) = \begin{cases}
{1}/{\sqrt{2a}} & \text{for } |x| \le a, \\[8pt]
0 & \text{for } |x|>a
\end{cases}</math>
then its Fourier transform is the sinc function,
<math display="block">\varphi(p)=\sqrt{\frac{a}{\pi \hbar}} \cdot \operatorname{sinc}\left(\frac{a p}{\hbar}\right)</math>
which yields infinite momentum variance despite having a centralized shape. The entropic uncertainty, on the other hand, is finite. Suppose for simplicity that the spatial resolution is just a two-bin measurement, ''δx''&nbsp;=&nbsp;''a'', and that the momentum resolution is ''δp''&nbsp;=&nbsp;''h''/''a''.

Partitioning the uniform spatial distribution into two equal bins is straightforward. We set the offset ''c''&nbsp;=&nbsp;1/2 so that the two bins span the distribution.
<math display="block">\operatorname P[x_0] = \int_{-a}^0 \frac{1}{2a} \, dx = \frac{1}{2}</math>
<math display="block">\operatorname P[x_1] = \int_0^a \frac{1}{2a} \, dx = \frac{1}{2}</math>
<math display="block">H_x = -\sum_{j=0}^{1} \operatorname P[x_j] \ln \operatorname P[x_j] = -\frac{1}{2} \ln \frac{1}{2} - \frac{1}{2} \ln \frac{1}{2} = \ln 2</math>

The bins for momentum must cover the entire real line. As done with the spatial distribution, we could apply an offset. It turns out, however, that the Shannon entropy is minimized when the zeroth bin for momentum is centered at the origin. (The reader is encouraged to try adding an offset.) The probability of lying within an arbitrary momentum bin can be expressed in terms of the [[sine integral]].

<math display="block">\begin{align}
\operatorname P[p_j] &= \frac{a}{\pi \hbar} \int_{(j-1/2)\delta p}^{(j+1/2)\delta p} \operatorname{sinc}^2\left(\frac{a p}{\hbar}\right) \, dp \\
&= \frac{1}{\pi} \int_{2\pi (j-1/2)}^{2\pi (j+1/2)} \operatorname{sinc}^2(u) \, du \\
&= \frac{1}{\pi} \left[ \operatorname {Si} ((4j+2)\pi)- \operatorname {Si} ((4j-2)\pi) \right]
\end{align}</math>

The Shannon entropy can be evaluated numerically.
<math display="block">H_p = -\sum_{j=-\infty}^\infty \operatorname P[p_j] \ln \operatorname P[p_j] = -\operatorname P[p_0] \ln \operatorname P[p_0]-2 \cdot \sum_{j=1}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] \approx 0.53</math>

The entropic uncertainty is indeed larger than the limiting value.
<math display="block">H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31</math>
|}

===Uncertainty relation with three angular momentum components===

For a particle of [[total angular momentum]] <math>j</math> the following uncertainty relation holds
<math display="block">
\sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j,
</math>
where <math>J_l</math> are angular momentum components. The relation can be derived from
<math display="block">
\langle J_x^2+J_y^2+J_z^2\rangle = j(j+1),
</math>
and
<math display="block">
\langle J_x\rangle^2+\langle J_y\rangle^2+\langle J_z\rangle^2\le j.
</math>
The relation can be strengthened as<ref name="PhysRevResearch21" /><ref>{{cite journal |last1=Chiew |first1=Shao-Hen |last2=Gessner |first2=Manuel |title=Improving sum uncertainty relations with the quantum Fisher information |journal=Physical Review Research |date=31 January 2022 |volume=4 |issue=1 |pages=013076 |doi=10.1103/PhysRevResearch.4.013076|arxiv=2109.06900 |bibcode=2022PhRvR...4a3076C |s2cid=237513883 }}</ref>
<math display="block">
\sigma_{J_x}^2+\sigma_{J_y}^2+F_Q[\varrho,J_z]/4\ge j,
</math>
where <math>F_Q[\varrho,J_z]</math> is the quantum Fisher information.

== History ==
{{See also|History of quantum mechanics}}
In 1925 Heisenberg published the [[Umdeutung paper|''Umdeutung'' (reinterpretation) paper]] where he showed that central aspect of quantum theory was the non-[[commutativity]]: the theory implied that the relative order of position and momentum measurement was significant. Working with [[Max Born]] and [[Pascual Jordan]], he continued to develop [[matrix mechanics]], that would become the first modern quantum mechanics formulation.<ref>{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity|volume= II: The modern theories, 1900–1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York|page=267}}</ref> 
[[File:Heisenbergbohr.jpg|thumb|Werner Heisenberg and Niels Bohr]]
In March 1926, working in Bohr's institute, Heisenberg realized that the non-[[commutativity]] implies the uncertainty principle. Writing to [[Wolfgang Pauli]] in February 1927, he worked out the basic concepts.<ref>{{Cite web |title=This Month in Physics History |url=http://www.aps.org/publications/apsnews/200802/physicshistory.cfm |access-date=2023-11-04 |website=www.aps.org |language=en |archive-date=2011-01-30 |archive-url=https://web.archive.org/web/20110130195156/http://aps.org/publications/apsnews/200802/physicshistory.cfm |url-status=live }}</ref>

In his celebrated 1927 paper "{{lang|de|Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik}}" ("On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics"), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement,<ref name=":0" /> but he did not give a precise definition for the uncertainties Δx and Δ''p''. Instead, he gave some plausible estimates in each case separately. His paper gave an analysis in terms of a microscope that Bohr showed was incorrect; Heisenberg included an addendum to the publication.

In his 1930 Chicago lecture<ref name="Heisenberg_1930">{{Citation |first=W. |last=Heisenberg |year=1930 |title=Physikalische Prinzipien der Quantentheorie |language=de|location=Leipzig |publisher=Hirzel }} English translation ''The Physical Principles of Quantum Theory''. Chicago: University of Chicago Press, 1930.</ref> he refined his principle:
{{NumBlk|:|<math>\Delta x \, \Delta p\gtrsim h</math>|{{EquationRef|A1}}}}

Later work broadened the concept. Any two variables that do not commute cannot be measured simultaneously—the more precisely one is known, the less precisely the other can be known. Heisenberg wrote:<blockquote>It can be expressed in its simplest form as follows: One can never know with perfect accuracy both of those two important factors which determine the movement of one of the smallest particles—its position and its velocity. It is impossible to determine accurately ''both'' the position and the direction and speed of a particle ''at the same instant''.<ref>Heisenberg, W., ''Die Physik der Atomkerne'', Taylor & Francis, 1952, p. 30.</ref></blockquote>

[[Earle Hesse Kennard|Kennard]]<ref name="Kennard" /><ref name=Sen2014 />{{rp|204}} in 1927 first proved the modern inequality:
{{NumBlk|:|<math>\sigma_x\sigma_p\ge\frac{\hbar}{2}</math>|{{EquationRef|A2}}}}
where {{math|1=''ħ'' = {{sfrac|''h''|2''π''}}}}, and {{math|''σ<sub>x</sub>''}}, {{math|''σ<sub>p</sub>''}} are the standard deviations of position and momentum. (Heisenberg only proved relation ({{EquationNote|A2}}) for the special case of Gaussian states.<ref name="Heisenberg_1930"/>) In 1929 Robertson generalized the inequality to all observables and in 1930 Schrödinger extended the form to allow non-zero covariance of the operators; this result is referred to as Robertson-Schrödinger inequality.<ref name=Sen2014 />{{rp|204}}

=== Terminology and translation ===
Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit",<ref name=":0" />
to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit". Later on, he always used "Unbestimmtheit". When the English-language version of Heisenberg's textbook, ''The Physical Principles of the Quantum Theory'', was published in 1930, however, only the English word "uncertainty" was used, and it became the term in the English language.<ref>{{Citation |first1=David |last1=Cassidy |year=2009 |title=Beyond Uncertainty: Heisenberg, Quantum Physics, and the Bomb |location= New York |publisher=Bellevue Literary Press |page=185 |bibcode=2010PhT....63a..49C |bibcode-access=free |last2=Saperstein |first2=Alvin M. |volume=63 |issue=1 |journal=Physics Today |doi=10.1063/1.3293416 |doi-access=free }}</ref>

=== Heisenberg's microscope ===
[[File:Heisenberg gamma ray microscope.svg|thumb|200px|right|Heisenberg's gamma-ray microscope for locating an electron (shown in blue). The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle ''θ''. The scattered gamma-ray is shown in red. Classical [[optics]] shows that the electron position can be resolved only up to an uncertainty Δ''x'' that depends on ''θ'' and the wavelength ''λ'' of the incoming light.]]
{{Main article|Heisenberg's microscope}}

The principle is quite counter-intuitive, so the early students of quantum theory had to be reassured that naive measurements to violate it were bound always to be unworkable. One way in which Heisenberg originally illustrated the intrinsic impossibility of violating the uncertainty principle is by using the [[observer effect (physics)|observer effect]] of an imaginary microscope as a measuring device.<ref name="Heisenberg_1930"/>

He imagines an experimenter trying to measure the position and momentum of an [[electron]] by shooting a [[photon]] at it.<ref name=GreensteinZajonc2006>{{cite book|first1=George |last1=Greenstein|first2=Arthur |last2=Zajonc|authorlink2=Arthur Zajonc|title=The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics|year=2006|publisher=Jones & Bartlett Learning|isbn=978-0-7637-2470-2}}</ref>{{rp|49–50}}
* Problem 1 – If the photon has a short [[wavelength]], and therefore, a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long [[wavelength]] and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
* Problem 2 – If a large [[aperture]] is used for the microscope, the electron's location can be well resolved (see [[Angular resolution#The_Rayleigh_criterion|Rayleigh criterion]]); but by the principle of [[conservation of momentum]], the transverse momentum of the incoming photon affects the electron's beamline momentum and hence, the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.

The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to the [[Planck constant]].<ref>{{Citation |last1=Tipler |first1=Paul A. |first2=Ralph A. |last2=Llewellyn |title=Modern Physics |volume=3 |publisher=W.H. Freeman & Co. |year=1999 |isbn=978-1572591646|lccn= 98046099
|url-access=|url=https://archive.org/details/modernphysics0003tipl |page=3 }}</ref> Heisenberg did not care to formulate the uncertainty principle as an exact limit, and preferred to use it instead, as a heuristic quantitative statement, correct up to small numerical factors, which makes the radically new noncommutativity of quantum mechanics inevitable.

===Intrinsic quantum uncertainty===

Historically, the uncertainty principle has been confused<ref>{{Citation|last=Furuta|first=Aya|title=One Thing Is Certain: Heisenberg's Uncertainty Principle Is Not Dead|journal=Scientific American|year=2012|url=https://www.scientificamerican.com/article/heisenbergs-uncertainty-principle-is-not-dead/|access-date=2018-10-20|archive-date=2022-04-01|archive-url=https://web.archive.org/web/20220401183444/https://www.scientificamerican.com/article/heisenbergs-uncertainty-principle-is-not-dead/|url-status=live}}</ref><ref name="Ozawa2003">{{Citation | last=Ozawa | first=Masanao | title=Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement | journal=Physical Review A | volume=67 | year=2003 | doi=10.1103/PhysRevA.67.042105|arxiv = quant-ph/0207121 |bibcode = 2003PhRvA..67d2105O | issue=4 | pages=42105 | s2cid=42012188}}</ref> with a related effect in [[physics]], called the [[observer effect (physics)|observer effect]], which notes that measurements of certain systems cannot be made without affecting the system,<ref>{{Citation |last=Wheeler |first=John Archibald |title=The 'Past' and the 'Delayed-Choice' Double-Slit Experiment |date=1978-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780124732506500066 |work=Mathematical Foundations of Quantum Theory |pages=9–48 |editor-last=Marlow |editor-first=A. R. |access-date=2023-07-19 |publisher=Academic Press |language=en |doi=10.1016/b978-0-12-473250-6.50006-6 |isbn=978-0-12-473250-6 |archive-date=2022-12-10 |archive-url=https://web.archive.org/web/20221210014455/https://www.sciencedirect.com/science/article/pii/B9780124732506500066 |url-status=live }}</ref><ref>{{Citation |last=Wheeler |first=John Archibald |title=Include the Observer in the Wave Function? |date=1977 |url=https://doi.org/10.1007/978-94-010-1196-9_1 |work=Quantum Mechanics, A Half Century Later: Papers of a Colloquium on Fifty Years of Quantum Mechanics, Held at the University Louis Pasteur, Strasbourg, May 2–4, 1974 |pages=1–18 |editor-last=Lopes |editor-first=José Leite |access-date=2023-07-19 |series=Episteme |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-010-1196-9_1 |isbn=978-94-010-1196-9 |editor2-last=Paty |editor2-first=Michel |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223170245/https://link.springer.com/chapter/10.1007/978-94-010-1196-9_1 |url-status=live }}</ref> that is, without changing something in a system. Heisenberg used such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.<ref>Werner Heisenberg, ''The Physical Principles of the Quantum Theory'', p. 20</ref> It has since become clearer, however, that the uncertainty principle is inherent in the properties of all [[wave|wave-like systems]],<ref name="Rozema">{{Cite journal | last1 = Rozema | first1 = L. A. | last2 = Darabi | first2 = A. | last3 = Mahler | first3 = D. H. | last4 = Hayat | first4 = A. | last5 = Soudagar | first5 = Y. | last6 = Steinberg | first6 = A. M. | doi = 10.1103/PhysRevLett.109.100404 |arxiv = 1208.0034v2| title = Violation of Heisenberg's Measurement–Disturbance Relationship by Weak Measurements | journal = Physical Review Letters | volume = 109 | issue = 10 | year = 2012 | pmid = 23005268|bibcode = 2012PhRvL.109j0404R | page=100404| s2cid = 37576344 }}</ref> and that it arises in quantum mechanics simply due to the [[matter wave]] nature of all quantum objects.<ref>{{Cite journal |last=De Broglie |first=Louis |date=October 1923 |title=Waves and Quanta |journal=Nature |language=en |volume=112 |issue=2815 |pages=540 |doi=10.1038/112540a0 |bibcode=1923Natur.112..540D |s2cid=186242764 |issn=1476-4687|doi-access=free }}</ref> Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.<ref name=nptel>{{YouTube|TcmGYe39XG0|Indian Institute of Technology Madras, Professor V. Balakrishnan, Lecture 1 – Introduction to Quantum Physics; Heisenberg's uncertainty principle, National Programme of Technology Enhanced Learning}}</ref>

== Critical reactions ==
{{Main article|Bohr–Einstein debates}}

The Copenhagen interpretation of quantum mechanics and Heisenberg's uncertainty principle were, in fact, initially seen as twin targets by detractors. According to the [[Copenhagen interpretation]] of quantum mechanics, there is no fundamental reality that the [[quantum state]] describes, just a prescription for calculating experimental results. There is no way to say what the state of a system fundamentally is, only what the result of observations might be.

[[Albert Einstein]] believed that randomness is a reflection of our ignorance of some fundamental property of reality, while [[Niels Bohr]] believed that the probability distributions are fundamental and irreducible, and depend on which measurements we choose to perform. [[Bohr–Einstein debates|Einstein and Bohr debated]] the uncertainty principle for many years.

=== Ideal detached observer ===
Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German):

{{Blockquote|"Like the moon has a definite position," Einstein said to me last winter, "whether or not we look at the moon, the same must also hold for the atomic objects, as there is no sharp distinction possible between these and macroscopic objects. Observation cannot ''create'' an element of reality like a position, there must be something contained in the complete description of physical reality which corresponds to the ''possibility'' of observing a position, already before the observation has been actually made." I hope, that I quoted Einstein correctly; it is always difficult to quote somebody out of memory with whom one does not agree. It is precisely this kind of postulate which I call the ideal of the detached observer.|Letter from Pauli to Niels Bohr, February 15, 1955<ref>{{cite book |last1=Enz |first1=Charles Paul |last2=von Meyenn |first2=Karl |title=Writings on Physics and Philosophy by Wolfgang Pauli |url=https://books.google.com/books?id=ueTd4g7pc5MC&pg=PA43 |publisher=Springer-Verlag |year=1994 |page=43 |translator=Robert Schlapp |isbn=3-540-56859-X |access-date=2018-02-10 |archive-date=2020-08-19 |archive-url=https://web.archive.org/web/20200819235529/https://books.google.com/books?id=ueTd4g7pc5MC&pg=PA43 |url-status=live }}</ref>}}

=== Einstein's slit ===
The first of Einstein's [[thought experiment]]s challenging the uncertainty principle went as follows:

{{quote|Consider a particle passing through a slit of width {{mvar|d}}. The slit introduces an uncertainty in momentum of approximately {{mvar|{{sfrac|h|d}}}} because the particle passes through the wall. But let us determine the momentum of the particle by measuring the recoil of the wall. In doing so, we find the momentum of the particle to arbitrary accuracy by conservation of momentum.}}

Bohr's response was that the wall is quantum mechanical as well, and that to measure the recoil to accuracy {{math|Δ''p''}}, the momentum of the wall must be known to this accuracy before the particle passes through. This introduces an uncertainty in the position of the wall and therefore the position of the slit equal to {{math|{{sfrac|''h''|Δ''p''}}}}, and if the wall's momentum is known precisely enough to measure the recoil, the slit's position is uncertain enough to disallow a position measurement.

A similar analysis with particles diffracting through multiple slits is given by [[Richard Feynman]].<ref>Feynman lectures on Physics, vol 3, 2–2</ref>

=== Einstein's box ===
Bohr was present when Einstein proposed the thought experiment which has become known as [[Einstein's box]]. Einstein argued that "Heisenberg's uncertainty equation implied that the uncertainty in time was related to the uncertainty in energy, the product of the two being related to the Planck constant."<ref name="Gamow">Gamow, G., ''The great physicists from Galileo to Einstein'', Courier Dover, 1988, p.260.</ref> Consider, he said, an ideal box, lined with mirrors so that it can contain light indefinitely. The box could be weighed before a clockwork mechanism opened an ideal shutter at a chosen instant to allow one single photon to escape. "We now know, explained Einstein, precisely the time at which the photon left the box."<ref>Kumar, M., ''Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality'', Icon, 2009, p. 282.</ref> "Now, weigh the box again. The change of mass tells the energy of the emitted light. In this manner, said Einstein, one could measure the energy emitted and the time it was released with any desired precision, in contradiction to the uncertainty principle."<ref name="Gamow" />

Bohr spent a sleepless night considering this argument, and eventually realized that it was flawed. He pointed out that if the box were to be weighed, say by a spring and a pointer on a scale, "since the box must move vertically with a change in its weight, there will be uncertainty in its vertical velocity and therefore an uncertainty in its height above the table. ... Furthermore, the uncertainty about the elevation above the Earth's surface will result in an uncertainty in the rate of the clock",<ref>Gamow, G., ''The great physicists from Galileo to Einstein'', Courier Dover, 1988, pp. 260–261. {{ISBN?}}</ref> because of Einstein's own theory of [[Gravitational time dilation|gravity's effect on time]]. "Through this chain of uncertainties, Bohr showed that Einstein's light box experiment could not simultaneously measure exactly both the energy of the photon and the time of its escape."<ref>{{cite book |last=Kumar |first=M. |title=Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality |publisher=Icon |year=2009 |page=287}}</ref>

=== EPR paradox for entangled particles ===
{{Main|Einstein–Podolsky–Rosen paradox}}
In 1935, Einstein, [[Boris Podolsky]] and [[Nathan Rosen]] published an analysis of spatially separated [[Quantum entanglement|entangled]] particles (EPR paradox).<ref>{{Cite journal |last1=Einstein |first1=A. |last2=Podolsky |first2=B. |last3=Rosen |first3=N. |date=1935-05-15 |title=Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? |journal=Physical Review |volume=47 |issue=10 |pages=777–780 |doi=10.1103/PhysRev.47.777|bibcode=1935PhRv...47..777E |doi-access=free }}</ref> According to EPR, one could measure the position of one of the entangled particles and the momentum of the second particle, and from those measurements deduce the position and momentum of both particles to any precision, violating the uncertainty principle. In order to avoid such possibility, the measurement of one particle must modify the probability distribution of the other particle instantaneously, possibly violating the [[principle of locality]].<ref>{{Cite book |last=Kumar |first=Manjit |title=Quantum: Einstein, Bohr and the great debate about the nature of reality |date=2011 |publisher=Norton |isbn=978-0-393-33988-8 |edition=1st paperback |location=New York}}</ref>

In 1964, [[John Stewart Bell]] showed that this assumption can be falsified, since it would imply a certain [[Bell's theorem|inequality]] between the probabilities of different experiments. [[Bell test|Experimental results]] confirm the predictions of quantum mechanics, ruling out EPR's basic assumption of [[Local hidden-variable theory|local hidden variables]].

=== Popper's criticism ===
{{Main article|Popper's experiment}}
Science philosopher [[Karl Popper]] approached the problem of indeterminacy as a logician and [[Philosophical realism|metaphysical realist]].<ref name="Popper1959">{{cite book | last1 = Popper | first1 = Karl | author-link1 = Karl Popper | title = The Logic of Scientific Discovery | publisher = Hutchinson & Co. | year = 1959| title-link = The Logic of Scientific Discovery }}</ref> He disagreed with the application of the uncertainty relations to individual particles rather than to [[Quantum ensemble|ensembles]] of identically prepared particles, referring to them as "statistical scatter relations".<ref name="Popper1959" /><ref name="Jarvie2006">{{cite book | last1 = Jarvie | first1 = Ian Charles | last2 = Milford | first2 = Karl | last3 = Miller | first3 = David W. | title = Karl Popper: a centenary assessment | volume = 3 | publisher = Ashgate | year = 2006 | isbn = 978-0-7546-5712-5}}</ref> In this statistical interpretation, a ''particular'' measurement may be made to arbitrary precision without invalidating the quantum theory.

In 1934, Popper published {{lang|de|italic=no|Zur Kritik der Ungenauigkeitsrelationen}} ("Critique of the Uncertainty Relations") in {{lang|de|[[Naturwissenschaften]]}},<ref name="Popper1934">{{cite journal | title = Zur Kritik der Ungenauigkeitsrelationen |language=de |trans-title=Critique of the Uncertainty Relations | journal = Naturwissenschaften | year = 1934 | first = Karl | last = Popper | author2 = Carl Friedrich von Weizsäcker | volume = 22 | issue = 48 | pages = 807–808 | doi=10.1007/BF01496543|bibcode = 1934NW.....22..807P | s2cid = 40843068}}</ref> and in the same year {{lang|de|[[The Logic of Scientific Discovery|Logik der Forschung]]}} (translated and updated by the author as ''The Logic of Scientific Discovery'' in 1959<ref name="Popper1959" />), outlining his arguments for the statistical interpretation. In 1982, he further developed his theory in ''Quantum theory and the schism in Physics'', writing:

{{quote|[Heisenberg's] formulae are, beyond all doubt, derivable ''statistical formulae'' of the quantum theory. But they have been ''habitually misinterpreted'' by those quantum theorists who said that these formulae can be interpreted as determining some upper limit to the ''precision of our measurements''. [original emphasis]<ref>{{cite book |last=Popper |first=K. |title=Quantum theory and the schism in Physics |publisher=Unwin Hyman |year=1982 |pages=53–54}}</ref>}}

Popper proposed an experiment to [[Falsifiability|falsify]] the uncertainty relations, although he later withdrew his initial version after discussions with [[Carl Friedrich von Weizsäcker]], Heisenberg, and Einstein; Popper sent his paper to Einstein and  it may have influenced the formulation of the EPR paradox.<ref name="Mehra2001">{{cite book | last1 = Mehra | first1 = Jagdish | last2 = Rechenberg | first2 = Helmut | author-link1 = Jagdish Mehra | author-link2 = Helmut Rechenberg | title = The Historical Development of Quantum Theory | publisher = Springer | year = 2001 | isbn = 978-0-387-95086-0 | url-access = registration | url = https://archive.org/details/completionofquan0000mehr }}</ref>{{rp|720}}

=== Free will ===
Some scientists, including [[Arthur Compton]]<ref>{{Cite journal | doi = 10.1126/science.74.1911.172| title = The Uncertainty Principle and Free Will| journal = Science| volume = 74| issue = 1911| pages = 172| year = 1931| last1 = Compton | first1 = A. H. | pmid=17808216|bibcode = 1931Sci....74..172C | s2cid = 29126625}}</ref> and [[Martin Heisenberg]],<ref>{{Cite journal | doi = 10.1038/459164a| pmid = 19444190| title = Is free will an illusion?| journal = Nature| volume = 459| issue = 7244| pages = 164–165| year = 2009| last1 = Heisenberg | first1 = M. |bibcode = 2009Natur.459..164H | s2cid = 4420023| doi-access = free}}</ref> have suggested that the uncertainty principle, or at least the general probabilistic nature of quantum mechanics, could be evidence for the two-stage model of free will. One critique, however, is that apart from the basic role of quantum mechanics as a foundation for chemistry, [[Quantum biology|nontrivial biological mechanisms requiring quantum mechanics]] are unlikely, due to the rapid [[Quantum decoherence|decoherence]] time of quantum systems at room temperature.<ref name="ReferenceA">{{Cite journal | doi = 10.1016/j.biosystems.2004.07.001| pmid = 15555759| title = Does quantum mechanics play a non-trivial role in life?| journal = Biosystems| volume = 78| issue = 1–3| pages = 69–79| year = 2004| last1 = Davies | first1 = P. C. W. | bibcode = 2004BiSys..78...69D}}</ref> Proponents of this theory commonly say that this decoherence is overcome by both screening and decoherence-free subspaces found in biological cells.<ref name="ReferenceA"/>

=== Thermodynamics ===
There is reason to believe that violating the uncertainty principle also strongly implies the violation of the [[second law of thermodynamics]].<ref>{{Cite journal |arxiv = 1205.6894|doi = 10.1038/ncomms2665|title = A violation of the uncertainty principle implies a violation of the second law of thermodynamics|year = 2013|last1 = Hänggi|first1 = Esther|last2 = Wehner|first2 = Stephanie|journal = Nature Communications|volume = 4|pages = 1670|pmid = 23575674|bibcode = 2013NatCo...4.1670H|s2cid = 205316392}}</ref> See [[Gibbs paradox]].

=== Rejection of the principle ===
Uncertainty principles relate quantum particles – electrons for example – to classical concepts – position and momentum. This presumes quantum particles have position and momentum. [[Edwin C. Kemble]] pointed out<ref>{{cite book |last=Kemble |first=E. C. |year=1937 |title=The Fundamental Principles of Quantum Mechanics |location=New York |publisher=McGraw-Hill, reprinted by Dover |page=244}}</ref>{{clarify inline|reason=What printing/edition does this page number refer to? Use year for that, and orig-year for original publication date|date=December 2024}} in 1937 that such properties cannot be experimentally verified and assuming they exist gives rise to many contradictions; similarly [[Rudolf Haag]] notes that position in quantum mechanics is an attribute of an interaction, say between an electron and a detector, not an intrinsic property.<ref>{{cite book |last=Haag |first=R. |year=1996 |title=Local Quantum Physics: Fields, Particles, Algebras |location=Berlin |publisher=Springer}}{{page?|date=February 2024}}{{ISBN?}}</ref><ref>{{Cite journal |last1=Peres |first1=Asher |url=https://link.aps.org/doi/10.1103/RevModPhys.76.93 |title=Quantum information and relativity theory |last2=Terno |first2=Daniel R. |journal=Reviews of Modern Physics |date=2004-01-06 |volume=76 |issue=1 |pages=93–123 [111] |language=en |doi=10.1103/RevModPhys.76.93 |arxiv=quant-ph/0212023 |bibcode=2004RvMP...76...93P |s2cid=7481797 |issn=0034-6861 |access-date=2024-01-25 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223160147/https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.76.93 |url-status=live }}</ref> From this point of view the uncertainty principle is not a fundamental quantum property but a concept "carried over from the language of our ancestors", as Kemble says.

==Applications==
Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. All forms of [[spectroscopy]], including [[particle physics]] use the relationship to relate measured energy line-width to the lifetime of quantum states. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. These include, for example, tests of number–phase uncertainty relations in [[superconductivity|superconducting]]<ref>{{cite journal|last1=Elion|first1=W. J. | last2=Matters | first2=M. | last3=Geigenmüller | first3=U. | last4=Mooij | first4=J. E. | title=Direct demonstration of Heisenberg's uncertainty principle in a superconductor | journal=Nature | volume=371 | pages=594–595 | year=1994 | doi= 10.1038/371594a0 | bibcode = 1994Natur.371..594E | issue=6498 | s2cid=4240085}}</ref> or [[quantum optics]]<ref>{{cite journal |last1=Smithey |first1=D. T. |first2=M. |last2=Beck |first3=J. |last3=Cooper |first4=M. G. |last4=Raymer | title=Measurement of number–phase uncertainty relations of optical fields | journal=Physical Review A |volume=48 | pages=3159–3167 | year=1993|doi=10.1103/PhysRevA.48.3159|bibcode = 1993PhRvA..48.3159S|issue=4|pmid=9909968}}</ref> systems. Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in [[gravitational-wave interferometer|gravitational wave interferometer]]s.<ref>{{cite journal|last=Caves|first=Carlton|title=Quantum-mechanical noise in an interferometer|journal=Physical Review D | volume=23 | pages=1693–1708 | year=1981|doi=10.1103/PhysRevD.23.1693|bibcode = 1981PhRvD..23.1693C|issue=8 }}</ref>

== See also ==
{{div col|colwidth=20em}}
* {{annotated link|Correspondence principle}}
* {{annotated link|Goodhart's law}} — when an attempt is made to use a statistical measure for purposes of control (directing), its statistical validity breaks down
* {{annotated link|Introduction to quantum mechanics}}
* {{annotated link|Küpfmüller's uncertainty principle}}
* {{annotated link|Quantum indeterminacy}}
* {{annotated link|Quantum superposition}} 
* {{annotated link|Quantum tunnelling}}
* ''{{annotated link|Physics and Beyond}}'' (Heisenberg's recollections)
* {{annotated link|Stronger uncertainty relations}}
{{div col end}}
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== Notes ==
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== References ==
{{reflist|30em}}

== External links ==
{{wikiquote}}
{{Commons category}}
* {{springer|title=Uncertainty principle|id=p/u095100}}
* [http://plato.stanford.edu/entries/qt-uncertainty/ Stanford Encyclopedia of Philosophy entry]

{{Quantum mechanics topics}}
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