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Uncertainty principle

From Wikipedia, the free encyclopedia
Foundational principle in quantum physics
For other uses, seeUncertainty principle (disambiguation).

Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
Canonical commutation rule for positionq and momentump variables of a particle, 1927.pq −qp =h/(2πi). Uncertainty principle of Heisenberg, 1927.

Theuncertainty principle, also known asHeisenberg's indeterminacy principle, is a fundamental concept inquantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position andmomentum, 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 ofmathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such asposition,x, and momentum,p.[1] Such paired-variables are known ascomplementary variables orcanonically conjugate variables.

First introduced in 1927 by German physicistWerner Heisenberg,[2][3][4][5] the formal inequality relating thestandard deviation of positionσx and the standard deviation of momentumσp was derived byEarle Hesse Kennard[6] later that year and byHermann Weyl[7] in 1928:

σxσp≥ℏ2{\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}}

whereℏ=h2π{\displaystyle \hbar ={\frac {h}{2\pi }}} is thereduced 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

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Main article:Introduction to quantum mechanics
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 arereal for illustrative purposes only; in quantum mechanics the wave function is generallycomplex.

It is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic[8] scales that humans experience. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. Thewave mechanics picture of the uncertainty principle is more visually intuitive, but the more abstractmatrix 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 correspondingorthonormalbases inHilbert space areFourier transforms of one another (i.e., position and momentum areconjugate variables). A nonzero function and its Fourier transform cannot both be sharply localized at the same time.[9] 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 asharp 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 thede Broglie relationp =ħk, wherek is thewavenumber.

Inmatrix mechanics, themathematical formulation of quantum mechanics, any pair of non-commutingself-adjoint operators representingobservables 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 observableA is performed, then the system is in a particular eigenstateΨ of that observable. However, the particular eigenstate of the observableA need not be an eigenstate of another observableB: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.[10]

Visualization

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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 areFourier transforms of each other: mathematically, the uncertainty principle expresses the relationship between conjugate variables in the transform.

Positionx and momentump wavefunctions corresponding to quantum particles. The colour opacity of the particles corresponds to theprobability density of finding the particle with positionx or momentum componentp.
Top: If wavelengthλ is unknown, so are momentump, wave-vectork and energyE (de Broglie relations). As the particle is more localized in position space, Δx is smaller than for Δpx.
Bottom: Ifλ is known, so arep,k, andE. As the particle is more localized in momentum space, Δp is smaller than for Δx.

Wave mechanics interpretation

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Main articles:Wave packet andSchrödinger equation
Propagation ofde Broglie waves in 1d—real part of thecomplex amplitude is blue, imaginary part is green. The probability (shown as the colouropacity) of finding the particle at a given pointx is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero thecurvature reverses sign, so the amplitude begins to decrease again, and vice versa—the result is an alternating amplitude: a wave.

According to thede Broglie hypothesis, every object in the universe is associated with awave. 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 wavenumberk0 or momentump0 is[11]ψ(x)∝eik0x=eip0x/ℏ .{\displaystyle \psi (x)\propto e^{ik_{0}x}=e^{ip_{0}x/\hbar }~.}

TheBorn rule states that this should be interpreted as aprobability density amplitude function in the sense that the probability of finding the particle betweena andb isP⁡[a≤X≤b]=∫ab|ψ(x)|2dx .{\displaystyle \operatorname {P} [a\leq X\leq b]=\int _{a}^{b}|\psi (x)|^{2}\,\mathrm {d} x~.}

In the case of the single-mode plane wave,|ψ(x)|2{\displaystyle |\psi (x)|^{2}} is1 ifX=x{\displaystyle X=x} and0 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 asum of many waves, which we may write asψ(x)∝∑nAneipnx/ℏ ,{\displaystyle \psi (x)\propto \sum _{n}A_{n}e^{ip_{n}x/\hbar }~,}whereAn represents the relative contribution of the modepn 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 thecontinuum limit, where the wave function is anintegral over all possible modesψ(x)=12πℏ∫−∞∞φ(p)⋅eipx/ℏdp ,{\displaystyle \psi (x)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\varphi (p)\cdot e^{ipx/\hbar }\,dp~,}withφ(p){\displaystyle \varphi (p)} representing the amplitude of these modes and is called the wave function inmomentum space. In mathematical terms, we say thatφ(p){\displaystyle \varphi (p)} is theFourier transform ofψ(x){\displaystyle \psi (x)} and thatx andp areconjugate 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.[12]

One way to quantify the precision of the position and momentum is thestandard deviation σ. Since|ψ(x)|2{\displaystyle |\psi (x)|^{2}} is a probability density function for position, we calculate its standard deviation.

The precision of the position is improved, i.e. reducedσx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increasedσp. Another way of stating this is thatσx andσp have aninverse 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

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We are interested in thevariances of position and momentum, defined asσx2=∫−∞∞x2⋅|ψ(x)|2dx−(∫−∞∞x⋅|ψ(x)|2dx)2{\displaystyle \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}}σp2=∫−∞∞p2⋅|φ(p)|2dp−(∫−∞∞p⋅|φ(p)|2dp)2 .{\displaystyle \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}~.}

Without loss of generality, we will assume that themeans 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 formσx2=∫−∞∞x2⋅|ψ(x)|2dx{\displaystyle \sigma _{x}^{2}=\int _{-\infty }^{\infty }x^{2}\cdot |\psi (x)|^{2}\,dx}σp2=∫−∞∞p2⋅|φ(p)|2dp .{\displaystyle \sigma _{p}^{2}=\int _{-\infty }^{\infty }p^{2}\cdot |\varphi (p)|^{2}\,dp~.}

The functionf(x)=x⋅ψ(x){\displaystyle f(x)=x\cdot \psi (x)} can be interpreted as avector in afunction space. We can define aninner product for a pair of functionsu(x) andv(x) in this vector space:⟨u∣v⟩=∫−∞∞u∗(x)⋅v(x)dx,{\displaystyle \langle u\mid v\rangle =\int _{-\infty }^{\infty }u^{*}(x)\cdot v(x)\,dx,}where the asterisk denotes thecomplex conjugate.

With this inner product defined, we note that the variance for position can be written asσx2=∫−∞∞|f(x)|2dx=⟨f∣f⟩ .{\displaystyle \sigma _{x}^{2}=\int _{-\infty }^{\infty }|f(x)|^{2}\,dx=\langle f\mid f\rangle ~.}

We can repeat this for momentum by interpreting the functiong~(p)=p⋅φ(p){\displaystyle {\tilde {g}}(p)=p\cdot \varphi (p)} as a vector, but we can also take advantage of the fact thatψ(x){\displaystyle \psi (x)} andφ(p){\displaystyle \varphi (p)} are Fourier transforms of each other. We evaluate the inverse Fourier transform throughintegration by parts:g(x)=12πℏ⋅∫−∞∞g~(p)⋅eipx/ℏdp=12πℏ∫−∞∞p⋅φ(p)⋅eipx/ℏdp=12πℏ∫−∞∞[p⋅∫−∞∞ψ(χ)e−ipχ/ℏdχ]⋅eipx/ℏdp=i2π∫−∞∞[ψ(χ)e−ipχ/ℏ|−∞∞−∫−∞∞dψ(χ)dχe−ipχ/ℏdχ]⋅eipx/ℏdp=−i∫−∞∞dψ(χ)dχ[12π∫−∞∞eip(x−χ)/ℏdp]dχ=−i∫−∞∞dψ(χ)dχ[δ(x−χℏ)]dχ=−iℏ∫−∞∞dψ(χ)dχ[δ(x−χ)]dχ=−iℏdψ(x)dx=(−iℏddx)⋅ψ(x),{\displaystyle {\begin{aligned}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{aligned}}}wherev=ℏ−ipe−ipχ/ℏ{\displaystyle v={\frac {\hbar }{-ip}}e^{-ip\chi /\hbar }} in the integration by parts, the cancelled term vanishes because the wave function vanishes at both infinities and|e−ipχ/ℏ|=1{\displaystyle |e^{-ip\chi /\hbar }|=1}, and then use theDirac delta function which is valid becausedψ(χ)dχ{\displaystyle {\dfrac {d\psi (\chi )}{d\chi }}} does not depend onp .

The term−iℏddx{\textstyle -i\hbar {\frac {d}{dx}}} is called themomentum operator in position space. ApplyingPlancherel's theorem, we see that the variance for momentum can be written asσp2=∫−∞∞|g~(p)|2dp=∫−∞∞|g(x)|2dx=⟨g∣g⟩.{\displaystyle \sigma _{p}^{2}=\int _{-\infty }^{\infty }|{\tilde {g}}(p)|^{2}\,dp=\int _{-\infty }^{\infty }|g(x)|^{2}\,dx=\langle g\mid g\rangle .}

TheCauchy–Schwarz inequality asserts thatσx2σp2=⟨f∣f⟩⋅⟨g∣g⟩≥|⟨f∣g⟩|2 .{\displaystyle \sigma _{x}^{2}\sigma _{p}^{2}=\langle f\mid f\rangle \cdot \langle g\mid g\rangle \geq |\langle f\mid g\rangle |^{2}~.}

Themodulus squared of any complex numberz can be expressed as|z|2=(Re(z))2+(Im(z))2≥(Im(z))2=(z−z∗2i)2.{\displaystyle |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}.}we letz=⟨f|g⟩{\displaystyle z=\langle f|g\rangle } andz∗=⟨g∣f⟩{\displaystyle z^{*}=\langle g\mid f\rangle } and substitute these into the equation above to get|⟨f∣g⟩|2≥(⟨f∣g⟩−⟨g∣f⟩2i)2 .{\displaystyle |\langle f\mid g\rangle |^{2}\geq \left({\frac {\langle f\mid g\rangle -\langle g\mid f\rangle }{2i}}\right)^{2}~.}

All that remains is to evaluate these inner products.

⟨f∣g⟩−⟨g∣f⟩=∫−∞∞ψ∗(x)x⋅(−iℏddx)ψ(x)dx−∫−∞∞ψ∗(x)(−iℏddx)⋅xψ(x)dx=iℏ⋅∫−∞∞ψ∗(x)[(−x⋅dψ(x)dx)+d(xψ(x))dx]dx=iℏ⋅∫−∞∞ψ∗(x)[(−x⋅dψ(x)dx)+ψ(x)+(x⋅dψ(x)dx)]dx=iℏ⋅∫−∞∞ψ∗(x)ψ(x)dx=iℏ⋅∫−∞∞|ψ(x)|2dx=iℏ{\displaystyle {\begin{aligned}\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{aligned}}}

Plugging this into the above inequalities, we getσx2σp2≥|⟨f∣g⟩|2≥(⟨f∣g⟩−⟨g∣f⟩2i)2=(iℏ2i)2=ℏ24{\displaystyle \sigma _{x}^{2}\sigma _{p}^{2}\geq |\langle f\mid g\rangle |^{2}\geq \left({\frac {\langle f\mid g\rangle -\langle g\mid f\rangle }{2i}}\right)^{2}=\left({\frac {i\hbar }{2i}}\right)^{2}={\frac {\hbar ^{2}}{4}}}and taking the square rootσxσp≥ℏ2 .{\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}~.}

with equality if and only ifp andx are linearly dependent. Note that the onlyphysics involved in this proof was thatψ(x){\displaystyle \psi (x)} andφ(p){\displaystyle \varphi (p)} are wave functions for position and momentum, which are Fourier transforms of each other. A similar result would hold forany pair of conjugate variables.

Matrix mechanics interpretation

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Main article:Matrix mechanics

In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators.[12] When considering pairs of observables, an important quantity is thecommutator. For a pair of operators andB^{\displaystyle {\hat {B}}}, one defines their commutator as[A^,B^]=A^B^−B^A^.{\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}.}In the case of position and momentum, the commutator is thecanonical commutation relation[x^,p^]=iℏ.{\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar .}

The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentumeigenstates. Let|ψ⟩{\displaystyle |\psi \rangle } be a right eigenstate of position with a constant eigenvaluex0. By definition, this means thatx^|ψ⟩=x0|ψ⟩.{\displaystyle {\hat {x}}|\psi \rangle =x_{0}|\psi \rangle .} Applying the commutator to|ψ⟩{\displaystyle |\psi \rangle } yields[x^,p^]|ψ⟩=(x^p^−p^x^)|ψ⟩=(x^−x0I^)p^|ψ⟩=iℏ|ψ⟩,{\displaystyle [{\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 ,}whereÎ is theidentity operator.

Suppose, for the sake ofproof by contradiction, that|ψ⟩{\displaystyle |\psi \rangle } is also a right eigenstate of momentum, with constant eigenvaluep0. If this were true, then one could write(x^−x0I^)p^|ψ⟩=(x^−x0I^)p0|ψ⟩=(x0I^−x0I^)p0|ψ⟩=0.{\displaystyle ({\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.}On the other hand, the above canonical commutation relation requires that[x^,p^]|ψ⟩=iℏ|ψ⟩≠0.{\displaystyle [{\hat {x}},{\hat {p}}]|\psi \rangle =i\hbar |\psi \rangle \neq 0.}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 isnot 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,σx=⟨x^2⟩−⟨x^⟩2{\displaystyle \sigma _{x}={\sqrt {\langle {\hat {x}}^{2}\rangle -\langle {\hat {x}}\rangle ^{2}}}}σp=⟨p^2⟩−⟨p^⟩2.{\displaystyle \sigma _{p}={\sqrt {\langle {\hat {p}}^{2}\rangle -\langle {\hat {p}}\rangle ^{2}}}.}

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

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Main articles:Quantum harmonic oscillator andStationary state

Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in terms of thecreation and annihilation operators:x^=ℏ2mω(a+a†){\displaystyle {\hat {x}}={\sqrt {\frac {\hbar }{2m\omega }}}(a+a^{\dagger })}p^=imωℏ2(a†−a).{\displaystyle {\hat {p}}=i{\sqrt {\frac {m\omega \hbar }{2}}}(a^{\dagger }-a).}

Using the standard rules for creation and annihilation operators on the energy eigenstates,a†|n⟩=n+1|n+1⟩{\displaystyle a^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle }a|n⟩=n|n−1⟩,{\displaystyle a|n\rangle ={\sqrt {n}}|n-1\rangle ,}the variances may be computed directly,σx2=ℏmω(n+12){\displaystyle \sigma _{x}^{2}={\frac {\hbar }{m\omega }}\left(n+{\frac {1}{2}}\right)}σp2=ℏmω(n+12).{\displaystyle \sigma _{p}^{2}=\hbar m\omega \left(n+{\frac {1}{2}}\right)\,.}The product of these standard deviations is thenσxσp=ℏ(n+12)≥ℏ2. {\displaystyle \sigma _{x}\sigma _{p}=\hbar \left(n+{\frac {1}{2}}\right)\geq {\frac {\hbar }{2}}.~}

In particular, the above Kennard bound[6] is saturated for theground staten=0, for which the probability density is just thenormal distribution.

Quantum harmonic oscillators with Gaussian initial condition

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Position (blue) and momentum (red) probability densities for an initial Gaussian distribution. From top to bottom, the animations show the casesΩ =ω,Ω = 2ω, andΩ =ω/2. Note the tradeoff between the widths of the distributions.

In a quantum harmonic oscillator of characteristic angular frequencyω, place a state that is offset from the bottom of the potential by some displacementx0 asψ(x)=(mΩπℏ)1/4exp⁡(−mΩ(x−x0)22ℏ),{\displaystyle \psi (x)=\left({\frac {m\Omega }{\pi \hbar }}\right)^{1/4}\exp {\left(-{\frac {m\Omega (x-x_{0})^{2}}{2\hbar }}\right)},}where Ω describes the width of the initial state but need not be the same asω. Through integration over thepropagator, we can solve for the full time-dependent solution. After many cancelations, the probability densities reduce to|Ψ(x,t)|2∼N(x0cos⁡(ωt),ℏ2mΩ(cos2⁡(ωt)+Ω2ω2sin2⁡(ωt))){\displaystyle |\Psi (x,t)|^{2}\sim {\mathcal {N}}\left(x_{0}\cos {(\omega t)},{\frac {\hbar }{2m\Omega }}\left(\cos ^{2}(\omega t)+{\frac {\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega t)}\right)\right)}|Φ(p,t)|2∼N(−mx0ωsin⁡(ωt),ℏmΩ2(cos2⁡(ωt)+ω2Ω2sin2⁡(ωt))),{\displaystyle |\Phi (p,t)|^{2}\sim {\mathcal {N}}\left(-mx_{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),}where we have used the notationN(μ,σ2){\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} to denote a normal distribution of meanμ and varianceσ2. Copying the variances above and applyingtrigonometric identities, we can write the product of the standard deviations asσxσp=ℏ2(cos2⁡(ωt)+Ω2ω2sin2⁡(ωt))(cos2⁡(ωt)+ω2Ω2sin2⁡(ωt))=ℏ43+12(Ω2ω2+ω2Ω2)−(12(Ω2ω2+ω2Ω2)−1)cos⁡(4ωt){\displaystyle {\begin{aligned}\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{aligned}}}

From the relationsΩ2ω2+ω2Ω2≥2,|cos⁡(4ωt)|≤1,{\displaystyle {\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\geq 2,\quad |\cos(4\omega t)|\leq 1,}we can conclude the following (the right most equality holds only whenΩ =ω):σxσp≥ℏ43+12(Ω2ω2+ω2Ω2)−(12(Ω2ω2+ω2Ω2)−1)=ℏ2.{\displaystyle \sigma _{x}\sigma _{p}\geq {\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}}.}

Coherent states

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Main article:Coherent state

A coherent state is a right eigenstate of theannihilation operator,a^|α⟩=α|α⟩,{\displaystyle {\hat {a}}|\alpha \rangle =\alpha |\alpha \rangle ,}which may be represented in terms ofFock states as|α⟩=e−|α|22∑n=0∞αnn!|n⟩{\displaystyle |\alpha \rangle =e^{-{|\alpha |^{2} \over 2}}\sum _{n=0}^{\infty }{\alpha ^{n} \over {\sqrt {n!}}}|n\rangle }

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,σx2=ℏ2mω,{\displaystyle \sigma _{x}^{2}={\frac {\hbar }{2m\omega }},}σp2=ℏmω2.{\displaystyle \sigma _{p}^{2}={\frac {\hbar m\omega }{2}}.}Therefore, every coherent state saturates the Kennard boundσxσp=ℏ2mωℏmω2=ℏ2.{\displaystyle \sigma _{x}\sigma _{p}={\sqrt {\frac {\hbar }{2m\omega }}}\,{\sqrt {\frac {\hbar m\omega }{2}}}={\frac {\hbar }{2}}.}with position and momentum each contributing an amountℏ/2{\textstyle {\sqrt {\hbar /2}}} in a "balanced" way. Moreover, everysqueezed 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

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Main article:Particle in a box

Consider a particle in a one-dimensional box of lengthL{\displaystyle L}. Theeigenfunctions in position and momentum space areψn(x,t)={Asin⁡(knx)e−iωnt,0<x<L,0,otherwise,{\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin(k_{n}x)\mathrm {e} ^{-\mathrm {i} \omega _{n}t},&0<x<L,\\0,&{\text{otherwise,}}\end{cases}}}andφn(p,t)=πLℏn(1−(−1)ne−ikL)e−iωntπ2n2−k2L2,{\displaystyle \varphi _{n}(p,t)={\sqrt {\frac {\pi L}{\hbar }}}\,\,{\frac {n\left(1-(-1)^{n}e^{-ikL}\right)e^{-i\omega _{n}t}}{\pi ^{2}n^{2}-k^{2}L^{2}}},}whereωn=π2ℏn28L2m{\textstyle \omega _{n}={\frac {\pi ^{2}\hbar n^{2}}{8L^{2}m}}} and we have used thede Broglie relationp=ℏk{\displaystyle p=\hbar k}. The variances ofx{\displaystyle x} andp{\displaystyle p} can be calculated explicitly:σx2=L212(1−6n2π2){\displaystyle \sigma _{x}^{2}={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)}σp2=(ℏnπL)2.{\displaystyle \sigma _{p}^{2}=\left({\frac {\hbar n\pi }{L}}\right)^{2}.}

The product of the standard deviations is thereforeσxσp=ℏ2n2π23−2.{\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}.}For alln=1,2,3,…{\displaystyle n=1,\,2,\,3,\,\ldots }, the quantityn2π23−2{\textstyle {\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}} is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs whenn=1{\displaystyle n=1}, in which caseσxσp=ℏ2π23−2≈0.568ℏ>ℏ2.{\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {\pi ^{2}}{3}}-2}}\approx 0.568\hbar >{\frac {\hbar }{2}}.}

Constant momentum

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Main article:Wave packet
Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space

Assume a particle initially has amomentum space wave function described by a normal distribution around some constant momentump0 according toφ(p)=(x0ℏπ)1/2exp⁡(−x02(p−p0)22ℏ2),{\displaystyle \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),}where we have introduced a reference scalex0=ℏ/mω0{\textstyle x_{0}={\sqrt {\hbar /m\omega _{0}}}}, withω0>0{\displaystyle \omega _{0}>0} 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 areΦ(p,t)=(x0ℏπ)1/2exp⁡(−x02(p−p0)22ℏ2−ip2t2mℏ),{\displaystyle \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),}Ψ(x,t)=(1x0π)1/2e−x02p02/2ℏ21+iω0texp⁡(−(x−ix02p0/ℏ)22x02(1+iω0t)).{\displaystyle \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).}

Since⟨p(t)⟩=p0{\displaystyle \langle p(t)\rangle =p_{0}} andσp(t)=ℏ/(2x0){\displaystyle \sigma _{p}(t)=\hbar /({\sqrt {2}}x_{0})}, 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 isσx=x021+ω02t2{\displaystyle \sigma _{x}={\frac {x_{0}}{\sqrt {2}}}{\sqrt {1+\omega _{0}^{2}t^{2}}}}such that the uncertainty product can only increase with time asσx(t)σp(t)=ℏ21+ω02t2{\displaystyle \sigma _{x}(t)\sigma _{p}(t)={\frac {\hbar }{2}}{\sqrt {1+\omega _{0}^{2}t^{2}}}}

Mathematical formalism

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Starting with Kennard's derivation of position-momentum uncertainty,Howard Percy Robertson developed[13][1] a formulation for arbitraryHermitian operatorsO^{\displaystyle {\hat {\mathcal {O}}}} expressed in terms of their standard deviationσO=⟨O^2⟩−⟨O^⟩2,{\displaystyle \sigma _{\mathcal {O}}={\sqrt {\langle {\hat {\mathcal {O}}}^{2}\rangle -\langle {\hat {\mathcal {O}}}\rangle ^{2}}},}where the brackets⟨O^⟩{\displaystyle \langle {\hat {\mathcal {O}}}\rangle } indicate anexpectation value of the observable represented by operatorO^{\displaystyle {\hat {\mathcal {O}}}}. For a pair of operatorsA^{\displaystyle {\hat {A}}} andB^{\displaystyle {\hat {B}}}, define their commutator as[A^,B^]=A^B^−B^A^,{\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}},}and the Robertson uncertainty relation is given by[14]σAσB≥|12i⟨[A^,B^]⟩|=12|⟨[A^,B^]⟩|.{\displaystyle \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|.}

Erwin Schrödinger[15] showed how to allow for correlation between the operators, giving a stronger inequality, known as theRobertson–Schrödinger uncertainty relation,[16][1]

σA2σB2≥|12⟨{A^,B^}⟩−⟨A^⟩⟨B^⟩|2+|12i⟨[A^,B^]⟩|2,{\displaystyle \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},}

where the anticommutator,{A^,B^}=A^B^+B^A^{\displaystyle \{{\hat {A}},{\hat {B}}\}={\hat {A}}{\hat {B}}+{\hat {B}}{\hat {A}}} is used.

Proof of theSchrödinger uncertainty relation

The derivation shown here incorporates and builds off of those shown in Robertson,[13] Schrödinger[16] and standard textbooks such as Griffiths.[17]: 138  For any Hermitian operatorA^{\displaystyle {\hat {A}}}, based upon the definition ofvariance, we haveσA2=⟨(A^−⟨A^⟩)Ψ|(A^−⟨A^⟩)Ψ⟩.{\displaystyle \sigma _{A}^{2}=\langle ({\hat {A}}-\langle {\hat {A}}\rangle )\Psi |({\hat {A}}-\langle {\hat {A}}\rangle )\Psi \rangle .}we let|f⟩=|(A^−⟨A^⟩)Ψ⟩{\displaystyle |f\rangle =|({\hat {A}}-\langle {\hat {A}}\rangle )\Psi \rangle } and thusσA2=⟨f∣f⟩.{\displaystyle \sigma _{A}^{2}=\langle f\mid f\rangle \,.}

Similarly, for any other Hermitian operatorB^{\displaystyle {\hat {B}}} in the same stateσB2=⟨(B^−⟨B^⟩)Ψ|(B^−⟨B^⟩)Ψ⟩=⟨g∣g⟩{\displaystyle \sigma _{B}^{2}=\langle ({\hat {B}}-\langle {\hat {B}}\rangle )\Psi |({\hat {B}}-\langle {\hat {B}}\rangle )\Psi \rangle =\langle g\mid g\rangle }for|g⟩=|(B^−⟨B^⟩)Ψ⟩.{\displaystyle |g\rangle =|({\hat {B}}-\langle {\hat {B}}\rangle )\Psi \rangle .}

The product of the two deviations can thus be expressed as

σA2σB2=⟨f∣f⟩⟨g∣g⟩.{\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}=\langle f\mid f\rangle \langle g\mid g\rangle .}1

In order to relate the two vectors|f⟩{\displaystyle |f\rangle } and|g⟩{\displaystyle |g\rangle }, we use theCauchy–Schwarz inequality[18] which is defined as⟨f∣f⟩⟨g∣g⟩≥|⟨f∣g⟩|2,{\displaystyle \langle f\mid f\rangle \langle g\mid g\rangle \geq |\langle f\mid g\rangle |^{2},}and thus Equation (1) can be written as

σA2σB2≥|⟨f∣g⟩|2.{\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}\geq |\langle f\mid g\rangle |^{2}.}2

Since⟨f∣g⟩{\displaystyle \langle f\mid g\rangle } is in general a complex number, we use the fact that the modulus squared of any complex numberz{\displaystyle z} is defined as|z|2=zz∗{\displaystyle |z|^{2}=zz^{*}}, wherez∗{\displaystyle z^{*}} is the complex conjugate ofz{\displaystyle z}. The modulus squared can also be expressed as

|z|2=(Re⁡(z))2+(Im⁡(z))2=(z+z∗2)2+(z−z∗2i)2.{\displaystyle |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}.}3

we letz=⟨f∣g⟩{\displaystyle z=\langle f\mid g\rangle } andz∗=⟨g∣f⟩{\displaystyle z^{*}=\langle g\mid f\rangle } and substitute these into the equation above to get

|⟨f∣g⟩|2=(⟨f∣g⟩+⟨g∣f⟩2)2+(⟨f∣g⟩−⟨g∣f⟩2i)2{\displaystyle |\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}}4

The inner product⟨f∣g⟩{\displaystyle \langle f\mid g\rangle } is written out explicitly as⟨f∣g⟩=⟨(A^−⟨A^⟩)Ψ|(B^−⟨B^⟩)Ψ⟩,{\displaystyle \langle f\mid g\rangle =\langle ({\hat {A}}-\langle {\hat {A}}\rangle )\Psi |({\hat {B}}-\langle {\hat {B}}\rangle )\Psi \rangle ,}and using the fact thatA^{\displaystyle {\hat {A}}} andB^{\displaystyle {\hat {B}}} are Hermitian operators, we find⟨f∣g⟩=⟨Ψ|(A^−⟨A^⟩)(B^−⟨B^⟩)Ψ⟩=⟨Ψ∣(A^B^−A^⟨B^⟩−B^⟨A^⟩+⟨A^⟩⟨B^⟩)Ψ⟩=⟨Ψ∣A^B^Ψ⟩−⟨Ψ∣A^⟨B^⟩Ψ⟩−⟨Ψ∣B^⟨A^⟩Ψ⟩+⟨Ψ∣⟨A^⟩⟨B^⟩Ψ⟩=⟨A^B^⟩−⟨A^⟩⟨B^⟩−⟨A^⟩⟨B^⟩+⟨A^⟩⟨B^⟩=⟨A^B^⟩−⟨A^⟩⟨B^⟩.{\displaystyle {\begin{aligned}\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{aligned}}}

Similarly it can be shown that⟨g∣f⟩=⟨B^A^⟩−⟨A^⟩⟨B^⟩.{\displaystyle \langle g\mid f\rangle =\langle {\hat {B}}{\hat {A}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle .}

Thus, we have⟨f∣g⟩−⟨g∣f⟩=⟨A^B^⟩−⟨A^⟩⟨B^⟩−⟨B^A^⟩+⟨A^⟩⟨B^⟩=⟨[A^,B^]⟩{\displaystyle \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 }and⟨f∣g⟩+⟨g∣f⟩=⟨A^B^⟩−⟨A^⟩⟨B^⟩+⟨B^A^⟩−⟨A^⟩⟨B^⟩=⟨{A^,B^}⟩−2⟨A^⟩⟨B^⟩.{\displaystyle \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 .}

We now substitute the above two equations above back into Eq. (4) and get|⟨f∣g⟩|2=(12⟨{A^,B^}⟩−⟨A^⟩⟨B^⟩)2+(12i⟨[A^,B^]⟩)2.{\displaystyle |\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}\,.}

Substituting the above into Equation (2) we get the Schrödinger uncertainty relationσAσB≥(12⟨{A^,B^}⟩−⟨A^⟩⟨B^⟩)2+(12i⟨[A^,B^]⟩)2.{\displaystyle \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}}}.}

This proof has an issue[19] related to the domains of the operators involved. For the proof to make sense, the vectorB^|Ψ⟩{\displaystyle {\hat {B}}|\Psi \rangle } has to be in the domain of theunbounded operatorA^{\displaystyle {\hat {A}}}, which is not always the case. In fact, the Robertson uncertainty relation is false ifA^{\displaystyle {\hat {A}}} is an angle variable andB^{\displaystyle {\hat {B}}} 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.[20] (See the counterexample section below.) This issue can be overcome by using avariational method for the proof,[21][22] or by working with an exponentiated version of the canonical commutation relations.[20]

Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operatorsA^{\displaystyle {\hat {A}}} andB^{\displaystyle {\hat {B}}} areself-adjoint operators. It suffices to assume that they are merelysymmetric operators. (The distinction between these two notions is generally glossed over in the physics literature, where the termHermitian is used for either or both classes of operators. See Chapter 9 of Hall's book[23] for a detailed discussion of this important but technical distinction.)

Phase space

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In thephase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given aWigner functionW(x,p){\displaystyle W(x,p)} withstar product ★ and a functionf, the following is generally true:[24]⟨f∗⋆f⟩=∫(f∗⋆f)W(x,p)dxdp≥0 .{\displaystyle \langle f^{*}\star f\rangle =\int (f^{*}\star f)\,W(x,p)\,dx\,dp\geq 0~.}

Choosingf=a+bx+cp{\displaystyle f=a+bx+cp}, we arrive at⟨f∗⋆f⟩=[a∗b∗c∗][1⟨x⟩⟨p⟩⟨x⟩⟨x⋆x⟩⟨x⋆p⟩⟨p⟩⟨p⋆x⟩⟨p⋆p⟩][abc]≥0 .{\displaystyle \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}}\geq 0~.}

Since this positivity condition is true foralla,b, andc, it follows that all the eigenvalues of the matrix are non-negative.

The non-negative eigenvalues then imply a corresponding non-negativity condition on thedeterminant,det[1⟨x⟩⟨p⟩⟨x⟩⟨x⋆x⟩⟨x⋆p⟩⟨p⟩⟨p⋆x⟩⟨p⋆p⟩]=det[1⟨x⟩⟨p⟩⟨x⟩⟨x2⟩⟨xp+iℏ2⟩⟨p⟩⟨xp−iℏ2⟩⟨p2⟩]≥0 ,{\displaystyle \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}}\geq 0~,}or, explicitly, after algebraic manipulation,σx2σp2=(⟨x2⟩−⟨x⟩2)(⟨p2⟩−⟨p⟩2)≥(⟨xp⟩−⟨x⟩⟨p⟩)2+ℏ24 .{\displaystyle \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)\geq \left(\langle xp\rangle -\langle x\rangle \langle p\rangle \right)^{2}+{\frac {\hbar ^{2}}{4}}~.}

Examples

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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.

Limitations

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The derivation of the Robertson inequality for operatorsA^{\displaystyle {\hat {A}}} andB^{\displaystyle {\hat {B}}} requiresA^B^ψ{\displaystyle {\hat {A}}{\hat {B}}\psi } andB^A^ψ{\displaystyle {\hat {B}}{\hat {A}}\psi } to be defined. There are quantum systems where these conditions are not valid.[27]One example is a quantumparticle on a ring, where the wave function depends on an angular variableθ{\displaystyle \theta } in the interval[0,2π]{\displaystyle [0,2\pi ]}. Define "position" and "momentum" operatorsA^{\displaystyle {\hat {A}}} andB^{\displaystyle {\hat {B}}} byA^ψ(θ)=θψ(θ),θ∈[0,2π],{\displaystyle {\hat {A}}\psi (\theta )=\theta \psi (\theta ),\quad \theta \in [0,2\pi ],}andB^ψ=−iℏdψdθ,{\displaystyle {\hat {B}}\psi =-i\hbar {\frac {d\psi }{d\theta }},}with periodic boundary conditions onB^{\displaystyle {\hat {B}}}. The definition ofA^{\displaystyle {\hat {A}}} depends theθ{\displaystyle \theta } range from 0 to2π{\displaystyle 2\pi }. These operators satisfy the usual commutation relations for position and momentum operators,[A^,B^]=iℏ{\displaystyle [{\hat {A}},{\hat {B}}]=i\hbar }. More precisely,A^B^ψ−B^A^ψ=iℏψ{\displaystyle {\hat {A}}{\hat {B}}\psi -{\hat {B}}{\hat {A}}\psi =i\hbar \psi } whenever bothA^B^ψ{\displaystyle {\hat {A}}{\hat {B}}\psi } andB^A^ψ{\displaystyle {\hat {B}}{\hat {A}}\psi } are defined, and the space of suchψ{\displaystyle \psi } is a dense subspace of the quantum Hilbert space.[28]

Now letψ{\displaystyle \psi } be any of the eigenstates ofB^{\displaystyle {\hat {B}}}, which are given byψ(θ)=e2πinθ{\displaystyle \psi (\theta )=e^{2\pi in\theta }}. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operatorA^{\displaystyle {\hat {A}}} is bounded, sinceθ{\displaystyle \theta } ranges over a bounded interval. Thus, in the stateψ{\displaystyle \psi }, the uncertainty ofB{\displaystyle B} is zero and the uncertainty ofA{\displaystyle A} is finite, so thatσAσB=0.{\displaystyle \sigma _{A}\sigma _{B}=0.}The Robertson uncertainty principle does not apply in this case:ψ{\displaystyle \psi } is not in the domain of the operatorB^A^{\displaystyle {\hat {B}}{\hat {A}}}, since multiplication byθ{\displaystyle \theta } disrupts the periodic boundary conditions imposed onB^{\displaystyle {\hat {B}}}.[20]

For the usual position and momentum operatorsX^{\displaystyle {\hat {X}}} andP^{\displaystyle {\hat {P}}} on the real line, no such counterexamples can occur. As long asσx{\displaystyle \sigma _{x}} andσp{\displaystyle \sigma _{p}} are defined in the stateψ{\displaystyle \psi }, the Heisenberg uncertainty principle holds, even ifψ{\displaystyle \psi } fails to be in the domain ofX^P^{\displaystyle {\hat {X}}{\hat {P}}} or ofP^X^{\displaystyle {\hat {P}}{\hat {X}}}.[29]

Mixed states

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The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all componentsϱk{\displaystyle \varrho _{k}} in any decomposition of thedensity matrix given asϱ=∑kpkϱk.{\displaystyle \varrho =\sum _{k}p_{k}\varrho _{k}.}Here, for the probabilitiespk≥0{\displaystyle p_{k}\geq 0} and∑kpk=1{\displaystyle \sum _{k}p_{k}=1} hold. Then, using the relation∑kak∑kbk≥(∑kakbk)2{\displaystyle \sum _{k}a_{k}\sum _{k}b_{k}\geq \left(\sum _{k}{\sqrt {a_{k}b_{k}}}\right)^{2}}forak,bk≥0{\displaystyle a_{k},b_{k}\geq 0},it follows that[30]σA2σB2≥[∑kpkL(ϱk)]2,{\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}\geq \left[\sum _{k}p_{k}L(\varrho _{k})\right]^{2},}where the function in the bound is definedL(ϱ)=|12tr⁡(ρ{A,B})−tr⁡(ρA)tr⁡(ρB)|2+|12itr⁡(ρ[A,B])|2.{\displaystyle 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}}}.}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 relationσA2σB2≥[maxpk,ϱk∑kpkL(ϱk)]2,{\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}\geq \left[\max _{p_{k},\varrho _{k}}\sum _{k}p_{k}L(\varrho _{k})\right]^{2},}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.[30]

With similar arguments, one can derive a relation with a convex roof on the right-hand side[30]σA2FQ[ϱ,B]≥4[minpk,Ψk∑kpkL(|Ψk⟩⟨Ψk|)]2{\displaystyle \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}}whereFQ[ϱ,B]{\displaystyle F_{Q}[\varrho ,B]} denotes thequantum Fisher information and the density matrix is decomposed to pure states asϱ=∑kpk|Ψk⟩⟨Ψk|.{\displaystyle \varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .}The derivation takes advantage of the fact that thequantum Fisher information is the convex roof of the variance times four.[31][32]

A simpler inequality follows without a convex roof[33]σA2FQ[ϱ,B]≥|⟨i[A,B]⟩|2,{\displaystyle \sigma _{A}^{2}F_{Q}[\varrho ,B]\geq \vert \langle i[A,B]\rangle \vert ^{2},}which is stronger than the Heisenberg uncertainty relation, since for the quantum Fisher information we haveFQ[ϱ,B]≤4σB,{\displaystyle F_{Q}[\varrho ,B]\leq 4\sigma _{B},}while for pure states the equality holds.

The Maccone–Pati uncertainty relations

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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 andArun K. Pati give non-trivial bounds on the sum of the variances for two incompatible observables.[34] (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref.[35] due to Yichen Huang.) For two non-commuting observablesA{\displaystyle A} andB{\displaystyle B} the first stronger uncertainty relation is given byσA2+σB2≥±i⟨Ψ∣[A,B]|Ψ⟩+∣⟨Ψ∣(A±iB)∣Ψ¯⟩|2,{\displaystyle \sigma _{A}^{2}+\sigma _{B}^{2}\geq \pm i\langle \Psi \mid [A,B]|\Psi \rangle +\mid \langle \Psi \mid (A\pm iB)\mid {\bar {\Psi }}\rangle |^{2},}whereσA2=⟨Ψ|A2|Ψ⟩−⟨Ψ∣A∣Ψ⟩2{\displaystyle \sigma _{A}^{2}=\langle \Psi |A^{2}|\Psi \rangle -\langle \Psi \mid A\mid \Psi \rangle ^{2}},σB2=⟨Ψ|B2|Ψ⟩−⟨Ψ∣B∣Ψ⟩2{\displaystyle \sigma _{B}^{2}=\langle \Psi |B^{2}|\Psi \rangle -\langle \Psi \mid B\mid \Psi \rangle ^{2}},|Ψ¯⟩{\displaystyle |{\bar {\Psi }}\rangle } is a normalized vector that is orthogonal to the state of the system|Ψ⟩{\displaystyle |\Psi \rangle } and one should choose the sign of±i⟨Ψ∣[A,B]∣Ψ⟩{\displaystyle \pm i\langle \Psi \mid [A,B]\mid \Psi \rangle } to make this real quantity a positive number.

The second stronger uncertainty relation is given byσA2+σB2≥12|⟨Ψ¯A+B∣(A+B)∣Ψ⟩|2{\displaystyle \sigma _{A}^{2}+\sigma _{B}^{2}\geq {\frac {1}{2}}|\langle {\bar {\Psi }}_{A+B}\mid (A+B)\mid \Psi \rangle |^{2}}where|Ψ¯A+B⟩{\displaystyle |{\bar {\Psi }}_{A+B}\rangle } is a state orthogonal to|Ψ⟩{\displaystyle |\Psi \rangle }.The form of|Ψ¯A+B⟩{\displaystyle |{\bar {\Psi }}_{A+B}\rangle } implies that the right-hand side of the new uncertainty relation is nonzero unless|Ψ⟩{\displaystyle |\Psi \rangle } is an eigenstate of(A+B){\displaystyle (A+B)}. One may note that|Ψ⟩{\displaystyle |\Psi \rangle } can be an eigenstate of(A+B){\displaystyle (A+B)} without being an eigenstate of eitherA{\displaystyle A} orB{\displaystyle B}. However, when|Ψ⟩{\displaystyle |\Psi \rangle } 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|Ψ⟩{\displaystyle |\Psi \rangle } is an eigenstate of both.

Energy–time

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An energy–time uncertainty relation likeΔEΔt≳ℏ/2,{\displaystyle \Delta E\Delta t\gtrsim \hbar /2,} has a long, controversial history; the meaning ofΔt{\displaystyle \Delta t} andΔE{\displaystyle \Delta E} varies and different formulations have different arenas of validity.[36] However, one well-known application is both well established[37][38] and experimentally verified:[39][40] the connection between the life-time of a resonance state,τ1/2{\displaystyle \tau _{\sqrt {1/2}}} and its energy widthΔE{\displaystyle \Delta E}:τ1/2ΔE=πℏ/4.{\displaystyle \tau _{\sqrt {1/2}}\Delta E=\pi \hbar /4.}In particle-physics, widths from experimental fits to theBreit–Wigner energy distribution are used to characterize the lifetime of quasi-stable or decaying states.[41]

An informal, heuristic meaning of the principle is the following:[42] 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, inspectroscopy, 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 thenatural linewidth. Fast-decaying states have a broad linewidth, while slow-decaying states have a narrow linewidth.[43] The same linewidth effect also makes it difficult to specify therest mass of unstable, fast-decaying particles inparticle physics. The faster theparticle decays (the shorter its lifetime), the less certain is its mass (the larger the particle'swidth).

Time in quantum mechanics

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The concept of "time" in quantum mechanics offers many challenges.[44] There is no quantum theory of time measurement; relativity is both fundamental to time and difficult to include in quantum mechanics.[36] 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.[1] The mathematical treatment of stable and unstable quantum systems differ.[45] These factors combine to make energy–time uncertainty principles controversial.

Three notions of "time" can be distinguished:[36] 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 accuracyΔE{\displaystyle \Delta E} requires a time intervalΔt>h/ΔE{\displaystyle \Delta t>h/\Delta E}.[38] However,Yakir Aharonov andDavid Bohm[46][36] 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".[44]: 31 

Observable time, measuring time between two events, remains a challenge for quantum theories; some progress has been made using positive operator-valued measure concepts.[36]

Mandelstam–Tamm

[edit]

In 1945,Leonid Mandelstam andIgor Tamm derived a non-relativistictime–energy uncertainty relation as follows.[47][36] From Heisenberg mechanics, the generalizedEhrenfest theorem for an observableB without explicit time dependence, represented by a self-adjoint operatorB^{\displaystyle {\hat {B}}} relates time dependence of the average value ofB^{\displaystyle {\hat {B}}} to the average of its commutator with the Hamiltonian:

d⟨B^⟩dt=iℏ⟨[H^,B^]⟩.{\displaystyle {\frac {d\langle {\hat {B}}\rangle }{dt}}={\frac {i}{\hbar }}\langle [{\hat {H}},{\hat {B}}]\rangle .}

The value of⟨[H^,B^]⟩{\displaystyle \langle [{\hat {H}},{\hat {B}}]\rangle } is then substituted in theRobertson uncertainty relation for the energy operatorH^{\displaystyle {\hat {H}}} andB^{\displaystyle {\hat {B}}}:σHσB≥|12i⟨[H^,B^]⟩|,{\displaystyle \sigma _{H}\sigma _{B}\geq \left|{\frac {1}{2i}}\langle [{\hat {H}},{\hat {B}}]\rangle \right|,}givingσHσB|d⟨B^⟩dt|≥ℏ2{\displaystyle \sigma _{H}{\frac {\sigma _{B}}{\left|{\frac {d\langle {\hat {B}}\rangle }{dt}}\right|}}\geq {\frac {\hbar }{2}}}(whenever the denominator is nonzero).While this is a universal result, it depends upon the observable chosen and that the deviationsσH{\displaystyle \sigma _{H}} andσB{\displaystyle \sigma _{B}} are computed for a particular state.IdentifyingΔE≡σE{\displaystyle \Delta E\equiv \sigma _{E}} and the characteristic timeτB≡σB|d⟨B^⟩dt|{\displaystyle \tau _{B}\equiv {\frac {\sigma _{B}}{\left|{\frac {d\langle {\hat {B}}\rangle }{dt}}\right|}}}gives an energy–time relationshipΔEτB≥ℏ2.{\displaystyle \Delta E\tau _{B}\geq {\frac {\hbar }{2}}.}AlthoughτB{\displaystyle \tau _{B}} has the dimension of time, it is different from the time parametert that enters theSchrödinger equation. ThisτB{\displaystyle \tau _{B}} can be interpreted as time for which the expectation value of the observable,⟨B^⟩,{\displaystyle \langle {\hat {B}}\rangle ,} changes by an amount equal to one standard deviation.[48]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:ΔT=ℏ/2ΔE.{\displaystyle \Delta T=\hbar /2\Delta E.} 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.[17]: 144 
  • ADelta particle, a quasistable composite of quarks related to protons and neutrons, has a lifetime of 10−23 s, so its measured mass equivalent to energy, 1232 MeV/c2, varies by ±120 MeV/c2; this variation is intrinsic and not caused by measurement errors.[17]: 144 
  • Two energy statesψ1,2{\displaystyle \psi _{1,2}} with energiesE1,2,{\displaystyle E_{1,2},} superimposed to create a composite state
Ψ(x,t)=aψ1(x)e−iE1t/h+bψ2(x)e−iE2t/h.{\displaystyle \Psi (x,t)=a\psi _{1}(x)e^{-iE_{1}t/h}+b\psi _{2}(x)e^{-iE_{2}t/h}.}
The probability amplitude of this state has a time-dependent interference term:
|Ψ(x,t)|2=a2|ψ1(x)|2+b2|ψ2(x)|2+2abcos⁡(E2−E1ℏt).{\displaystyle |\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).}
The oscillation period varies inversely with the energy difference:τ=2πℏ/(E2−E1){\displaystyle \tau =2\pi \hbar /(E_{2}-E_{1})}.[17]: 144 

Each example has a different meaning for the time uncertainty, according to the observable and state used.

Quantum field theory

[edit]

Some formulations ofquantum field theory uses temporary electron–positron pairs in its calculations calledvirtual 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 ofall 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 violateconservation 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.[17]: 145  The energy of the universe is not an exactly known parameter at all times.[1] When events transpire at very short time intervals, there is uncertainty in the energy of these events.

Harmonic analysis

[edit]
Main article:Fourier transform § Uncertainty principle

In the context ofharmonic 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,(∫−∞∞x2|f(x)|2dx)(∫−∞∞ξ2|f^(ξ)|2dξ)≥‖f‖2416π2.{\displaystyle \left(\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }\xi ^{2}|{\hat {f}}(\xi )|^{2}\,d\xi \right)\geq {\frac {\|f\|_{2}^{4}}{16\pi ^{2}}}.}

Further mathematical uncertainty inequalities, including the aboveentropic uncertainty, hold between a functionf and its Fourier transform ƒ̂:[49][50][51]Hx+Hξ≥log⁡(e/2){\displaystyle H_{x}+H_{\xi }\geq \log(e/2)}

Signal processing

[edit]

In the context oftime–frequency analysis uncertainty principles are referred to as theGabor limit, afterDennis Gabor, or sometimes theHeisenberg–Gabor limit. The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be bothtime limited andband limited (a function and its Fourier transform cannot both have bounded domain)—seebandlimited versus timelimited. More accurately, thetime-bandwidth orduration-bandwidth product satisfiesσtσf≥14π≈0.08 cycles,{\displaystyle \sigma _{t}\sigma _{f}\geq {\frac {1}{4\pi }}\approx 0.08{\text{ cycles}},}whereσt{\displaystyle \sigma _{t}} andσf{\displaystyle \sigma _{f}} are the standard deviations of the time and frequency energy concentrations respectively.[52] The minimum is attained for aGaussian-shaped pulse (Gabor wavelet) [For the un-squared Gaussian (i.e. signal amplitude) and its un-squared Fourier transform magnitudeσtσf=1/2π{\displaystyle \sigma _{t}\sigma _{f}=1/2\pi }; squaring reduces eachσ{\displaystyle \sigma } by a factor2{\displaystyle {\sqrt {2}}}.] Another common measure is the product of the time and frequencyfull width at half maximum (of the power/energy), which for the Gaussian equals2ln⁡2/π≈0.44{\displaystyle 2\ln 2/\pi \approx 0.44} (seebandwidth-limited pulse).

Stated differently, one cannot simultaneously sharply localize a signalf in both thetime domain andfrequency domain.

When applied tofilters, the result implies that one cannot simultaneously achieve a high temporal resolution and high frequency resolution at the same time; a concrete example are theresolution 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 thesimultaneous 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 thetransients are important, thewavelet transform is often used instead of the Fourier.

Discrete Fourier transform

[edit]

Let{xn}:=x0,x1,…,xN−1{\displaystyle \left\{\mathbf {x_{n}} \right\}:=x_{0},x_{1},\ldots ,x_{N-1}} be a sequence ofN complex numbers and{Xk}:=X0,X1,…,XN−1,{\displaystyle \left\{\mathbf {X_{k}} \right\}:=X_{0},X_{1},\ldots ,X_{N-1},} be its discrete Fourier transform.

Denote by‖x‖0{\displaystyle \|x\|_{0}} the number of non-zero elements in the time sequencex0,x1,…,xN−1{\displaystyle x_{0},x_{1},\ldots ,x_{N-1}} and by‖X‖0{\displaystyle \|X\|_{0}} the number of non-zero elements in the frequency sequenceX0,X1,…,XN−1{\displaystyle X_{0},X_{1},\ldots ,X_{N-1}}. Then,‖x‖0⋅‖X‖0≥N.{\displaystyle \|x\|_{0}\cdot \|X\|_{0}\geq N.}

This inequality issharp, with equality achieved whenx orX is a Dirac mass, or more generally whenx is a nonzero multiple of a Dirac comb supported on a subgroup of the integers moduloN (in which caseX is also a Dirac comb supported on a complementary subgroup, and vice versa).

More generally, ifT andW are subsets of the integers moduloN, letLT,RW:ℓ2(Z/NZ)→ℓ2(Z/NZ){\displaystyle L_{T},R_{W}:\ell ^{2}(\mathbb {Z} /N\mathbb {Z} )\to \ell ^{2}(\mathbb {Z} /N\mathbb {Z} )} denote the time-limiting operator andband-limiting operators, respectively. Then‖LTRW‖2≤|T||W||G|{\displaystyle \|L_{T}R_{W}\|^{2}\leq {\frac {|T||W|}{|G|}}}where the norm is theoperator norm of operators on the Hilbert spaceℓ2(Z/NZ){\displaystyle \ell ^{2}(\mathbb {Z} /N\mathbb {Z} )} of functions on the integers moduloN. This inequality has implications forsignal reconstruction.[53]

WhenN is aprime number, a stronger inequality holds:‖x‖0+‖X‖0≥N+1.{\displaystyle \|x\|_{0}+\|X\|_{0}\geq N+1.}Discovered byTerence Tao, this inequality is also sharp.[54]

Benedicks's theorem

[edit]

Amrein–Berthier[55] and Benedicks's theorem[56] intuitively says that the set of points wheref is non-zero and the set of points whereƒ̂ is non-zero cannot both be small.

Specifically, it is impossible for a functionf inL2(R) and its Fourier transformƒ̂ to both besupported on sets of finiteLebesgue measure. A more quantitative version is[57][58]‖f‖L2(Rd)≤CeC|S||Σ|(‖f‖L2(Sc)+‖f^‖L2(Σc)) .{\displaystyle \|f\|_{L^{2}(\mathbf {R} ^{d})}\leq Ce^{C|S||\Sigma |}{\bigl (}\|f\|_{L^{2}(S^{c})}+\|{\hat {f}}\|_{L^{2}(\Sigma ^{c})}{\bigr )}~.}

One expects that the factorCeC|S||Σ| may be replaced byCeC(|S||Σ|)1/d, which is only known if eitherS orΣ is convex.

Hardy's uncertainty principle

[edit]

The mathematicianG. H. Hardy formulated the following uncertainty principle:[59] it is not possible forf and ƒ̂ to both be "very rapidly decreasing". Specifically, iff inL2(R){\displaystyle L^{2}(\mathbb {R} )} is such that|f(x)|≤C(1+|x|)Ne−aπx2{\displaystyle |f(x)|\leq C(1+|x|)^{N}e^{-a\pi x^{2}}}and|f^(ξ)|≤C(1+|ξ|)Ne−bπξ2{\displaystyle |{\hat {f}}(\xi )|\leq C(1+|\xi |)^{N}e^{-b\pi \xi ^{2}}} (C>0,N{\displaystyle C>0,N} an integer),then, ifab > 1,f = 0, while ifab = 1, then there is a polynomialP of degree≤N such thatf(x)=P(x)e−aπx2.{\displaystyle f(x)=P(x)e^{-a\pi x^{2}}.}

This was later improved as follows: iff∈L2(Rd){\displaystyle f\in L^{2}(\mathbb {R} ^{d})} is such that∫Rd∫Rd|f(x)||f^(ξ)|eπ|⟨x,ξ⟩|(1+|x|+|ξ|)Ndxdξ<+∞ ,{\displaystyle \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 ~,}thenf(x)=P(x)e−π⟨Ax,x⟩ ,{\displaystyle f(x)=P(x)e^{-\pi \langle Ax,x\rangle }~,}whereP is a polynomial of degree(N −d)/2 andA is a reald ×d positive definite matrix.

This result was stated in Beurling's complete works without proof and proved in Hörmander[60] (the cased=1,N=0{\displaystyle d=1,N=0}) and Bonami, Demange, and Jaming[61] for the general case. Note that Hörmander–Beurling's version implies the caseab > 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.[62]

A full description of the caseab < 1 as well as the following extension to Schwartz class distributions appears in ref.[63]

Theorem— If a tempered distributionf∈S′(Rd){\displaystyle f\in {\mathcal {S}}'(\mathbb {R} ^{d})} is such thateπ|x|2f∈S′(Rd){\displaystyle e^{\pi |x|^{2}}f\in {\mathcal {S}}'(\mathbb {R} ^{d})}andeπ|ξ|2f^∈S′(Rd) ,{\displaystyle e^{\pi |\xi |^{2}}{\hat {f}}\in {\mathcal {S}}'(\mathbb {R} ^{d})~,}thenf(x)=P(x)e−π⟨Ax,x⟩ ,{\displaystyle f(x)=P(x)e^{-\pi \langle Ax,x\rangle }~,}for some convenient polynomialP and real positive definite matrixA of typed ×d.

Additional uncertainty relations

[edit]

Heisenberg limit

[edit]

Inquantum metrology, and especiallyinterferometry, theHeisenberg 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 abeam-splitter) and the energy is given by the number of photons used in aninterferometer. Although some claim to have broken the Heisenberg limit, this reflects disagreement on the definition of the scaling resource.[64] 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.[65]

Systematic and statistical errors

[edit]

The inequalities above focus on thestatistical imprecision of observables as quantified by the standard deviationσ{\displaystyle \sigma }. Heisenberg's original version, however, was dealing with thesystematic error, a disturbance of the quantum system produced by the measuring apparatus, i.e., an observer effect.

If we letεA{\displaystyle \varepsilon _{A}} represent the error (i.e.,inaccuracy) of a measurement of an observableA andηB{\displaystyle \eta _{B}} the disturbance produced on a subsequent measurement of the conjugate variableB by the former measurement ofA, then the inequality proposed by Masanao Ozawa − encompassing both systematic and statistical errors - holds:[66]

εAηB+εAσB+σAηB≥12|⟨[A^,B^]⟩|{\displaystyle \varepsilon _{A}\,\eta _{B}+\varepsilon _{A}\,\sigma _{B}+\sigma _{A}\,\eta _{B}\,\geq \,{\frac {1}{2}}\,\left|{\Bigl \langle }{\bigl [}{\hat {A}},{\hat {B}}{\bigr ]}{\Bigr \rangle }\right|}

Heisenberg's uncertainty principle, as originally described in the 1927 formulation, mentions only the first term of Ozawa inequality, regarding thesystematic error. Using the notation above to describe theerror/disturbance effect ofsequential measurements (firstA, thenB), it could be written as

εAηB≥12|⟨[A^,B^]⟩|{\displaystyle \varepsilon _{A}\,\eta _{B}\,\geq \,{\frac {1}{2}}\,\left|{\Bigl \langle }{\bigl [}{\hat {A}},{\hat {B}}{\bigr ]}{\Bigr \rangle }\right|}

The formal derivation of the Heisenberg relation is possible but far from intuitive. It wasnot proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years.[67][68]Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errorsσA{\displaystyle \sigma _{A}} andσB{\displaystyle \sigma _{B}}. There is increasing experimental evidence[69][70][71][72] 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,[1] it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case ofsimultaneous measurements (A andB at the same time):

εAεB≥12|⟨[A^,B^]⟩|{\displaystyle \varepsilon _{A}\,\varepsilon _{B}\,\geq \,{\frac {1}{2}}\,\left|{\Bigl \langle }{\bigl [}{\hat {A}},{\hat {B}}{\bigr ]}{\Bigr \rangle }\right|}

The two simultaneous measurements onA andB are necessarily[73]unsharp orweak.

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[1]

σAσB≥12|⟨[A^,B^]⟩|{\displaystyle \sigma _{A}\,\sigma _{B}\,\geq \,{\frac {1}{2}}\,\left|{\Bigl \langle }{\bigl [}{\hat {A}},{\hat {B}}{\bigr ]}{\Bigr \rangle }\right|}

and Ozawa relations we obtainεAηB+εAσB+σAηB+σAσB≥|⟨[A^,B^]⟩|.{\displaystyle \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|.}The four terms can be written as:(εA+σA)(ηB+σB)≥|⟨[A^,B^]⟩|.{\displaystyle (\varepsilon _{A}+\sigma _{A})\,(\eta _{B}+\sigma _{B})\,\geq \,\left|{\Bigl \langle }{\bigl [}{\hat {A}},{\hat {B}}{\bigr ]}{\Bigr \rangle }\right|.}Defining:ε¯A≡(εA+σA){\displaystyle {\bar {\varepsilon }}_{A}\,\equiv \,(\varepsilon _{A}+\sigma _{A})}as theinaccuracy in the measured values of the variableA andη¯B≡(ηB+σB){\displaystyle {\bar {\eta }}_{B}\,\equiv \,(\eta _{B}+\sigma _{B})}as theresulting fluctuation in the conjugate variableB, Kazuo Fujikawa[74] established an uncertainty relation similar to the Heisenberg original one, but valid both forsystematic and statistical errors:

ε¯Aη¯B≥|⟨[A^,B^]⟩|{\displaystyle {\bar {\varepsilon }}_{A}\,{\bar {\eta }}_{B}\,\geq \,\left|{\Bigl \langle }{\bigl [}{\hat {A}},{\hat {B}}{\bigr ]}{\Bigr \rangle }\right|}

Quantum entropic uncertainty principle

[edit]

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.[22][75][76][77] Other examples include highlybimodal distributions, orunimodal distributions with divergent variance.

A solution that overcomes these issues is an uncertainty based onentropic uncertainty instead of the product of variances. While formulating themany-worlds interpretation of quantum mechanics in 1957,Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty.[78] This conjecture, also studied by I. I. Hirschman[79] and proven in 1975 by W. Beckner[80] and by Iwo Bialynicki-Birula and Jerzy Mycielski[81] is that, for two normalized, dimensionless Fourier transform pairsf(a) andg(b) where

f(a)=∫−∞∞g(b) e2πiabdb{\displaystyle f(a)=\int _{-\infty }^{\infty }g(b)\ e^{2\pi iab}\,db}    and   g(b)=∫−∞∞f(a) e−2πiabda{\displaystyle \,\,\,g(b)=\int _{-\infty }^{\infty }f(a)\ e^{-2\pi iab}\,da}

the Shannoninformation entropiesHa=−∫−∞∞|f(a)|2log⁡|f(a)|2da,{\displaystyle H_{a}=-\int _{-\infty }^{\infty }|f(a)|^{2}\log |f(a)|^{2}\,da,}andHb=−∫−∞∞|g(b)|2log⁡|g(b)|2db{\displaystyle H_{b}=-\int _{-\infty }^{\infty }|g(b)|^{2}\log |g(b)|^{2}\,db}are subject to the following constraint,

Ha+Hb≥log⁡(e/2){\displaystyle H_{a}+H_{b}\geq \log(e/2)}

where the logarithms may be in any base.

The probability distribution functions associated with the position wave functionψ(x) and the momentum wave functionφ(x) have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless byHx=−∫|ψ(x)|2ln⁡(x0|ψ(x)|2)dx=−⟨ln⁡(x0|ψ(x)|2)⟩{\displaystyle 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 }Hp=−∫|φ(p)|2ln⁡(p0|φ(p)|2)dp=−⟨ln⁡(p0|φ(p)|2)⟩{\displaystyle 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 }wherex0 andp0 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 theFourier transform relation between the position wave functionψ(x) and the momentum wavefunctionφ(p), the above constraint can be written for the corresponding entropies as

Hx+Hp≥log⁡(eh2x0p0){\displaystyle H_{x}+H_{p}\geq \log \left({\frac {e\,h}{2\,x_{0}\,p_{0}}}\right)}

whereh is thePlanck constant.

Depending on one's choice of thex0 p0 product, the expression may be written in many ways. Ifx0p0 is chosen to beh, thenHx+Hp≥log⁡(e2){\displaystyle H_{x}+H_{p}\geq \log \left({\frac {e}{2}}\right)}

If, instead,x0p0 is chosen to beħ, thenHx+Hp≥log⁡(eπ){\displaystyle H_{x}+H_{p}\geq \log(e\,\pi )}

Ifx0 andp0 are chosen to be unity in whatever system of units are being used, thenHx+Hp≥log⁡(eh2){\displaystyle H_{x}+H_{p}\geq \log \left({\frac {e\,h}{2}}\right)}whereh 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.[82]

The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverselogarithmic Sobolev inequalities[83]Hx≤12log⁡(2eπσx2/x02) ,{\displaystyle H_{x}\leq {\frac {1}{2}}\log(2e\pi \sigma _{x}^{2}/x_{0}^{2})~,}Hp≤12log⁡(2eπσp2/p02) ,{\displaystyle H_{p}\leq {\frac {1}{2}}\log(2e\pi \sigma _{p}^{2}/p_{0}^{2})~,}(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 isstronger than the one based on standard deviations, becauseσxσp≥ℏ2exp⁡(Hx+Hp−log⁡(eh2x0p0))≥ℏ2 .{\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}\exp \left(H_{x}+H_{p}-\log \left({\frac {e\,h}{2\,x_{0}\,p_{0}}}\right)\right)\geq {\frac {\hbar }{2}}~.}

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 ofbase 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 theShannon entropy has been used,not the quantumvon Neumann entropy. Finally, the normal distribution saturates the inequality, and it is the only distribution with this property, because it is themaximum entropy probability distribution among those with fixed variance (cf.here for proof).

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

x0=ℏ2mω{\displaystyle x_{0}={\sqrt {\frac {\hbar }{2m\omega }}}}ψ(x)=(mωπℏ)1/4exp⁡(−mωx22ℏ)=(12πx02)1/4exp⁡(−x24x02){\displaystyle {\begin{aligned}\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{aligned}}}

The probability distribution is the normal distribution|ψ(x)|2=1x02πexp⁡(−x22x02){\displaystyle |\psi (x)|^{2}={\frac {1}{x_{0}{\sqrt {2\pi }}}}\exp {\left(-{\frac {x^{2}}{2x_{0}^{2}}}\right)}}with Shannon entropyHx=−∫|ψ(x)|2ln⁡(|ψ(x)|2⋅x0)dx=−1x02π∫−∞∞exp⁡(−x22x02)ln⁡[12πexp⁡(−x22x02)]dx=12π∫−∞∞exp⁡(−u22)[ln⁡(2π)+u22]du=ln⁡(2π)+12.{\displaystyle {\begin{aligned}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{aligned}}}

A completely analogous calculation proceeds for the momentum distribution. Choosing a standard momentum ofp0=ℏ/x0{\displaystyle p_{0}=\hbar /x_{0}}:φ(p)=(2x02πℏ2)1/4exp⁡(−x02p2ℏ2){\displaystyle \varphi (p)=\left({\frac {2x_{0}^{2}}{\pi \hbar ^{2}}}\right)^{1/4}\exp {\left(-{\frac {x_{0}^{2}p^{2}}{\hbar ^{2}}}\right)}}|φ(p)|2=2x02πℏ2exp⁡(−2x02p2ℏ2){\displaystyle |\varphi (p)|^{2}={\sqrt {\frac {2x_{0}^{2}}{\pi \hbar ^{2}}}}\exp {\left(-{\frac {2x_{0}^{2}p^{2}}{\hbar ^{2}}}\right)}}Hp=−∫|φ(p)|2ln⁡(|φ(p)|2⋅ℏ/x0)dp=−2x02πℏ2∫−∞∞exp⁡(−2x02p2ℏ2)ln⁡[2πexp⁡(−2x02p2ℏ2)]dp=2π∫−∞∞exp⁡(−2v2)[ln⁡(π2)+2v2]dv=ln⁡(π2)+12.{\displaystyle {\begin{aligned}H_{p}&=-\int |\varphi (p)|^{2}\ln(|\varphi (p)|^{2}\cdot \hbar /x_{0})\,dp\\&=-{\sqrt {\frac {2x_{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{aligned}}}

The entropic uncertainty is therefore the limiting valueHx+Hp=ln⁡(2π)+12+ln⁡(π2)+12=1+ln⁡π=ln⁡(eπ).{\displaystyle {\begin{aligned}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{aligned}}}

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. Letδx be a measure of the spatial resolution. We take the zeroth bin to be centered near the origin, with possibly some small constant offsetc. The probability of lying within the jth interval of widthδx isP⁡[xj]=∫(j−1/2)δx−c(j+1/2)δx−c|ψ(x)|2dx{\displaystyle \operatorname {P} [x_{j}]=\int _{(j-1/2)\delta x-c}^{(j+1/2)\delta x-c}|\psi (x)|^{2}\,dx}

To account for this discretization, we can define the Shannon entropy of the wave function for a given measurement apparatus asHx=−∑j=−∞∞P⁡[xj]ln⁡P⁡[xj].{\displaystyle H_{x}=-\sum _{j=-\infty }^{\infty }\operatorname {P} [x_{j}]\ln \operatorname {P} [x_{j}].}

Under the above definition, the entropic uncertainty relation isHx+Hp>ln⁡(e2)−ln⁡(δxδph).{\displaystyle H_{x}+H_{p}>\ln \left({\frac {e}{2}}\right)-\ln \left({\frac {\delta x\delta p}{h}}\right).}

Here we note thatδxδp/h is a typical infinitesimal phase space volume used in the calculation of apartition function. The inequality is also strict and not saturated. Efforts to improve this bound are an active area of research.

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.

ψ(x)=(mωπℏ)1/4exp⁡(−mωx22ℏ){\displaystyle \psi (x)=\left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\exp {\left(-{\frac {m\omega x^{2}}{2\hbar }}\right)}}

The probability of lying within one of these bins can be expressed in terms of theerror function.

P⁡[xj]=mωπℏ∫(j−1/2)δx(j+1/2)δxexp⁡(−mωx2ℏ)dx=1π∫(j−1/2)δxmω/ℏ(j+1/2)δxmω/ℏeu2du=12[erf⁡((j+12)δx⋅mωℏ)−erf⁡((j−12)δx⋅mωℏ)]{\displaystyle {\begin{aligned}\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{aligned}}}

The momentum probabilities are completely analogous.

P⁡[pj]=12[erf⁡((j+12)δp⋅1ℏmω)−erf⁡((j−12)δx⋅1ℏmω)]{\displaystyle \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]}

For simplicity, we will set the resolutions toδx=hmω{\displaystyle \delta x={\sqrt {\frac {h}{m\omega }}}}δp=hmω{\displaystyle \delta p={\sqrt {hm\omega }}}so that the probabilities reduce toP⁡[xj]=P⁡[pj]=12[erf⁡((j+12)2π)−erf⁡((j−12)2π)]{\displaystyle \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]}

The Shannon entropy can be evaluated numerically.Hx=Hp=−∑j=−∞∞P⁡[xj]ln⁡P⁡[xj]=−∑j=−∞∞12[erf⁡((j+12)2π)−erf⁡((j−12)2π)]ln⁡12[erf⁡((j+12)2π)−erf⁡((j−12)2π)]≈0.3226{\displaystyle {\begin{aligned}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{aligned}}}

The entropic uncertainty is indeed larger than the limiting value.Hx+Hp≈0.3226+0.3226=0.6452>ln⁡(e2)−ln⁡1≈0.3069{\displaystyle H_{x}+H_{p}\approx 0.3226+0.3226=0.6452>\ln \left({\frac {e}{2}}\right)-\ln 1\approx 0.3069}

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

Sinc function example
An example of a unimodal distribution with infinite variance is thesinc function. If the wave function is the correctly normalized uniform distribution,

ψ(x)={1/2afor |x|≤a,0for |x|>a{\displaystyle \psi (x)={\begin{cases}{1}/{\sqrt {2a}}&{\text{for }}|x|\leq a,\\[8pt]0&{\text{for }}|x|>a\end{cases}}}then its Fourier transform is the sinc function,φ(p)=aπℏ⋅sinc⁡(apℏ){\displaystyle \varphi (p)={\sqrt {\frac {a}{\pi \hbar }}}\cdot \operatorname {sinc} \left({\frac {ap}{\hbar }}\right)}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 = a, and that the momentum resolution isδp = h/a.

Partitioning the uniform spatial distribution into two equal bins is straightforward. We set the offsetc = 1/2 so that the two bins span the distribution.P⁡[x0]=∫−a012adx=12{\displaystyle \operatorname {P} [x_{0}]=\int _{-a}^{0}{\frac {1}{2a}}\,dx={\frac {1}{2}}}P⁡[x1]=∫0a12adx=12{\displaystyle \operatorname {P} [x_{1}]=\int _{0}^{a}{\frac {1}{2a}}\,dx={\frac {1}{2}}}Hx=−∑j=01P⁡[xj]ln⁡P⁡[xj]=−12ln⁡12−12ln⁡12=ln⁡2{\displaystyle 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}

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 thesine integral.

P⁡[pj]=aπℏ∫(j−1/2)δp(j+1/2)δpsinc2⁡(apℏ)dp=1π∫2π(j−1/2)2π(j+1/2)sinc2⁡(u)du=1π[Si⁡((4j+2)π)−Si⁡((4j−2)π)]{\displaystyle {\begin{aligned}\operatorname {P} [p_{j}]&={\frac {a}{\pi \hbar }}\int _{(j-1/2)\delta p}^{(j+1/2)\delta p}\operatorname {sinc} ^{2}\left({\frac {ap}{\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{aligned}}}

The Shannon entropy can be evaluated numerically.Hp=−∑j=−∞∞P⁡[pj]ln⁡P⁡[pj]=−P⁡[p0]ln⁡P⁡[p0]−2⋅∑j=1∞P⁡[pj]ln⁡P⁡[pj]≈0.53{\displaystyle 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}

The entropic uncertainty is indeed larger than the limiting value.Hx+Hp≈0.69+0.53=1.22>ln⁡(e2)−ln⁡1≈0.31{\displaystyle H_{x}+H_{p}\approx 0.69+0.53=1.22>\ln \left({\frac {e}{2}}\right)-\ln 1\approx 0.31}

Uncertainty relation with three angular momentum components

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For a particle oftotal angular momentumj{\displaystyle j} the following uncertainty relation holdsσJx2+σJy2+σJz2≥j,{\displaystyle \sigma _{J_{x}}^{2}+\sigma _{J_{y}}^{2}+\sigma _{J_{z}}^{2}\geq j,}whereJl{\displaystyle J_{l}} are angular momentum components. The relation can be derived from⟨Jx2+Jy2+Jz2⟩=j(j+1),{\displaystyle \langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\rangle =j(j+1),}and⟨Jx⟩2+⟨Jy⟩2+⟨Jz⟩2≤j.{\displaystyle \langle J_{x}\rangle ^{2}+\langle J_{y}\rangle ^{2}+\langle J_{z}\rangle ^{2}\leq j.}The relation can be strengthened as[30][84]σJx2+σJy2+FQ[ϱ,Jz]/4≥j,{\displaystyle \sigma _{J_{x}}^{2}+\sigma _{J_{y}}^{2}+F_{Q}[\varrho ,J_{z}]/4\geq j,}whereFQ[ϱ,Jz]{\displaystyle F_{Q}[\varrho ,J_{z}]} is the quantum Fisher information.

History

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See also:History of quantum mechanics

In 1925 Heisenberg published theUmdeutung (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 withMax Born andPascual Jordan, he continued to developmatrix mechanics, that would become the first modern quantum mechanics formulation.[85]

Werner Heisenberg and Niels Bohr

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. Writing toWolfgang Pauli in February 1927, he worked out the basic concepts.[86]

In his celebrated 1927 paper "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,[2] but he did not give a precise definition for the uncertainties Δx and Δp. 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[87] he refined his principle:

ΔxΔp≳h{\displaystyle \Delta x\,\Delta p\gtrsim h}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:

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 accuratelyboth the position and the direction and speed of a particleat the same instant.[88]

Kennard[6][1]: 204  in 1927 first proved the modern inequality:

σxσp≥ℏ2{\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}}A2

whereħ =⁠h/2π⁠, andσx,σp are the standard deviations of position and momentum. (Heisenberg only proved relation (A2) for the special case of Gaussian states.[87]) 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.[1]: 204 

Terminology and translation

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Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit",[2]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.[89]

Heisenberg's microscope

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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 angleθ. The scattered gamma-ray is shown in red. Classicaloptics shows that the electron position can be resolved only up to an uncertainty Δx that depends onθ and the wavelengthλ 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 theobserver effect of an imaginary microscope as a measuring device.[87]

He imagines an experimenter trying to measure the position and momentum of anelectron by shooting aphoton at it.[90]: 49–50 

  • Problem 1 – If the photon has a shortwavelength, 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 longwavelength 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 largeaperture is used for the microscope, the electron's location can be well resolved (seeRayleigh criterion); but by the principle ofconservation 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 thePlanck constant.[91] 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

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Historically, the uncertainty principle has been confused[92][66] with a related effect inphysics, called theobserver effect, which notes that measurements of certain systems cannot be made without affecting the system,[93][94] 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.[95] It has since become clearer, however, that the uncertainty principle is inherent in the properties of allwave-like systems,[69] and that it arises in quantum mechanics simply due to thematter wave nature of all quantum objects.[96] Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.[97]

Critical reactions

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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 theCopenhagen interpretation of quantum mechanics, there is no fundamental reality that thequantum 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, whileNiels Bohr believed that the probability distributions are fundamental and irreducible, and depend on which measurements we choose to perform.Einstein and Bohr debated the uncertainty principle for many years.

Ideal detached observer

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

"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 cannotcreate an element of reality like a position, there must be something contained in the complete description of physical reality which corresponds to thepossibility 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[98]

Einstein's slit

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The first of Einstein'sthought experiments challenging the uncertainty principle went as follows:

Consider a particle passing through a slit of widthd. The slit introduces an uncertainty in momentum of approximately⁠h/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 accuracyΔp, 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 to⁠h/Δp⁠, 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 byRichard Feynman.[99]

Einstein's box

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Bohr was present when Einstein proposed the thought experiment which has become known asEinstein'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."[100] 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."[101] "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."[100]

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",[102] because of Einstein's own theory ofgravity'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."[103]

EPR paradox for entangled particles

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Main article:Einstein–Podolsky–Rosen paradox

In 1935, Einstein,Boris Podolsky andNathan Rosen published an analysis of spatially separatedentangled particles (EPR paradox).[104] 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 theprinciple of locality.[105]

In 1964,John Stewart Bell showed that this assumption can be falsified, since it would imply a certaininequality between the probabilities of different experiments.Experimental results confirm the predictions of quantum mechanics, ruling out EPR's basic assumption oflocal hidden variables.

Popper's criticism

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Main article:Popper's experiment

Science philosopherKarl Popper approached the problem of indeterminacy as a logician andmetaphysical realist.[106] He disagreed with the application of the uncertainty relations to individual particles rather than toensembles of identically prepared particles, referring to them as "statistical scatter relations".[106][107] In this statistical interpretation, aparticular measurement may be made to arbitrary precision without invalidating the quantum theory.

In 1934, Popper publishedZur Kritik der Ungenauigkeitsrelationen ("Critique of the Uncertainty Relations") inNaturwissenschaften,[108] and in the same yearLogik der Forschung (translated and updated by the author asThe Logic of Scientific Discovery in 1959[106]), outlining his arguments for the statistical interpretation. In 1982, he further developed his theory inQuantum theory and the schism in Physics, writing:

[Heisenberg's] formulae are, beyond all doubt, derivablestatistical formulae of the quantum theory. But they have beenhabitually misinterpreted by those quantum theorists who said that these formulae can be interpreted as determining some upper limit to theprecision of our measurements. [original emphasis][109]

Popper proposed an experiment tofalsify the uncertainty relations, although he later withdrew his initial version after discussions withCarl Friedrich von Weizsäcker, Heisenberg, and Einstein; Popper sent his paper to Einstein and it may have influenced the formulation of the EPR paradox.[110]: 720 

Free will

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Some scientists, includingArthur Compton[111] andMartin Heisenberg,[112] 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,nontrivial biological mechanisms requiring quantum mechanics are unlikely, due to the rapiddecoherence time of quantum systems at room temperature.[113] Proponents of this theory commonly say that this decoherence is overcome by both screening and decoherence-free subspaces found in biological cells.[113]

Thermodynamics

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There is reason to believe that violating the uncertainty principle also strongly implies the violation of thesecond law of thermodynamics.[114] SeeGibbs paradox.

Rejection of the principle

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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[115][clarification needed] in 1937 that such properties cannot be experimentally verified and assuming they exist gives rise to many contradictions; similarlyRudolf Haag notes that position in quantum mechanics is an attribute of an interaction, say between an electron and a detector, not an intrinsic property.[116][117] 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

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Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. All forms ofspectroscopy, includingparticle 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 insuperconducting[118] orquantum optics[119] systems. Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required ingravitational wave interferometers.[120]

See also

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References

[edit]
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