|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Other uses}}
| | Hello from France. I'm glad to be here. My first name is Roseanne. <br>I live in a town called Vertou in nothern France.<br>I was also born in Vertou 35 years ago. Married in January 2000. I'm working at the university.<br><br>my web site: [http://www.premierpour.com/weddingwire-brides-choice-2011/ fifa 15 coin generator] |
| {{Quantum mechanics}}
| |
| In [[quantum mechanics]], the '''uncertainty principle''' is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as [[Position (vector)|position]] ''x'' and [[momentum]] ''p'', can be known simultaneously. For instance, in 1927, [[Werner Heisenberg]] stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.<ref name="Heisenberg_1927">{{Citation |first=W. |last=Heisenberg |title={{lang|de|Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik}} |journal=[[Zeitschrift für Physik]] |volume=43 |issue=3–4 |year=1927 |pages=172–198 |doi=10.1007/BF01397280 |postscript=. |bibcode = 1927ZPhy...43..172H }}.
| |
| Annotated pre-publication proof sheet of [http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/papers/corr155.1.html Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik], March 23, 1927.</ref> The formal inequality relating the [[standard deviation]] of position σ<sub>x</sub> and the standard deviation of momentum σ<sub>p</sub> was derived by [[Earle Hesse Kennard]]<ref name="Kennard">{{Citation |first=E. H. |last=Kennard |title={{lang|de|Zur Quantenmechanik einfacher Bewegungstypen}} |journal=Zeitschrift für Physik |volume=44 |issue=4–5 |year=1927 |pages=326 |doi=10.1007/BF01391200 |postscript=. |bibcode = 1927ZPhy...44..326K }}</ref> later that year and by [[Hermann Weyl]]<ref name="Weyl1928">{{Citation|last=Weyl|first=H.|title=Gruppentheorie und Quantenmechanik|year=1928|publisher=Hirzel|location=Leipzig}}</ref> in 1928,
| |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math> \sigma_{x}\sigma_{p} \geq \frac{\hbar}{2}, </math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| where ''ħ'' is the reduced [[Planck constant]].
| |
| | |
| The original heuristic argument that such a limit should exist was given by Heisenberg, after whom it is sometimes named the '''Heisenberg principle'''. This ascribes the uncertainty in the measurable quantities to the jolt-like disturbance triggered by the act of observation. Though widely repeated in textbooks, this physical argument is now known to be fundamentally misleading.<ref>[http://prl.aps.org/abstract/PRL/v109/i10/e100404 Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements Rozema L A et al. ''Phys. Rev. Lett''. '''109''', 100404 (2012)]</ref><ref>[http://www.sciencedaily.com/releases/2012/09/120907125154.htm Scientists Cast Doubt On Heisenberg's Uncertainty Principle ''Science Daily'' 7 September 2012]</ref> While the act of measurement does lead to uncertainty, the loss of precision is less than that predicted by Heisenberg's argument; the formal mathematical result remains valid, however.
| |
| | |
| Historically, the uncertainty principle has been confused<ref>{{Citation|last=Furuta|first=Aya|title=One Thing Is Certain: Heisenberg's Uncertainty Principle Is Not Dead|journal=Scientific American|year=2012|url=http://www.scientificamerican.com/article.cfm?id=heisenbergs-uncertainty-principle-is-not-dead}}</ref><ref name="Ozawa2003">{{Citation|last=Ozawa|first=Masanao|title=Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement|journal=Physical Review A|volume=67|year=2003|doi=10.1103/PhysRevA.67.042105|arxiv = quant-ph/0207121 |bibcode = 2003PhRvA..67d2105O|issue=4 }}</ref> with a somewhat similar effect in [[physics]], called the [[observer effect (physics)|observer effect]], which notes that measurements of certain systems cannot be made without affecting the systems. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.<ref>Werner Heisenberg, ''The Physical Principles of the Quantum Theory'', p. 20</ref> It has since become clear, however, that the uncertainty principle is inherent in the properties of all [[wave|wave-like systems]],<ref>{{Cite doi|10.1103/PhysRevLett.109.100404}}</ref> and that it arises in quantum mechanics simply due to the [[matter wave]] nature of all quantum objects. Thus, ''the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology''.<ref name=nptel>[http://www.youtube.com/watch?v=TcmGYe39XG0 youtube.com website] Indian Institute of Technology Madras, Professor V. Balakrishnan, Lecture 1 – Introduction to Quantum Physics; Heisenberg's uncertainty principle, National Programme of Technology Enhanced Learning</ref> It must be emphasized that ''measurement'' does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.<ref>Quantum Mechanics Non-Relativistic Theory, Third Edition: Volume 3. Landau, Lifshitz</ref>
| |
| | |
| Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. These include, for example, tests of number-phase uncertainty relations in [[superconductivity|superconducting]]<ref>{{Citation|last=Elion|first=W. J.|coauthors=M. Matters, U. Geigenmüller & J. E. Mooij|title=Direct demonstration of Heisenberg's uncertainty principle in a superconductor|journal=Nature|volume=371|pages=594–595|year=1994|doi=10.1038/371594a0|bibcode = 1994Natur.371..594E|issue=6498 }}</ref> or [[quantum optics]]<ref>{{Citation|last=Smithey|first=D. T.|coauthors=M. Beck, J. Cooper, M. G. Raymer|title=Measurement of number-phase uncertainty relations of optical fields|journal=Phys. Rev. A|volume=48|pages=3159–3167|year=1993|doi=10.1103/PhysRevA.48.3159|bibcode = 1993PhRvA..48.3159S|issue=4|pmid=9909968 }}</ref> systems. Applications dependent on the uncertainty principle for their operation include extremely low noise technology such as that required in [[gravitational-wave interferometer]]s.<ref>{{Citation|last=Caves|first=Carlton|title=Quantum-mechanical noise in an interferometer|journal=Phys. Rev. D|volume=23|pages=1693–1708|year=1981|doi=10.1103/PhysRevD.23.1693|bibcode = 1981PhRvD..23.1693C|issue=8 }}</ref>
| |
| | |
| ==Introduction==
| |
| {{Main|Introduction to quantum mechanics}}
| |
| As a principle, Heisenberg's uncertainty relationship must be something that is in accord with all experience. However, humans do not form an intuitive understanding of this indeterminacy in everyday life, so it may be helpful to demonstrate how it is integral to more easily understood physical situations. Two alternative conceptualizations of quantum physics can be examined with the goal of demonstrating the key role the uncertainty principle plays. A [[Schrödinger equation|wave mechanics]] picture of the uncertainty principle provides for a more visually intuitive demonstration, and the somewhat more abstract [[matrix mechanics]] picture provides for a demonstration of the uncertainty principle that is more easily generalized to cover a multitude of physical contexts.
| |
| | |
| Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding [[basis (linear algebra)|bases]] are [[Fourier transforms]] of one another (i.e., position and momentum are [[conjugate variables]]). A nonzero function and its Fourier transform cannot both be sharply localized. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a [[Dirac delta function|sharp spike]] at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of the particle takes the form of a [[matter wave]], and momentum is its Fourier conjugate, assured by the de Broglie relation {{math|''p'' {{=}} ''ħk''}}, where {{mvar|k}} is the [[wavenumber]].
| |
| | |
| In matrix mechanics the [[mathematical formulation of quantum mechanics#Postulates of quantum mechanics|mathematical formulation of quantum mechanics]], any pair of non-[[commutator|commuting]] [[self-adjoint operator]]s representing [[observable]]s are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable {{mvar|A}} is performed, then the system is in a particular eigenstate {{mvar|Ψ}} of that observable. However, the particular eigenstate of the observable {{mvar|A}} need not be an eigenstate of another observable {{mvar|B}}: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.<ref>{{Citation|author=Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë|title=Quantum mechanics|year=1996|publisher=Wiley|location=Wiley-Interscience|isbn=978-0-471-56952-7|pages=231–233}}</ref>
| |
| | |
| ===Wave mechanics interpretation===
| |
| [[File:Uncertainty principle.gif|360px|"360px"|right|thumb| Click to see animation. The evolution of an initially very localized gaussian wave function of a free particle in two-dimensional space, with colour and intensity indicating phase and amplitude. The spreading of the wave function in all directions shows that the initial momentum has a spread of values, unmodified in time; while the spread in position increases in time: as a result, the uncertainty ''Δx Δp'' increases in time.]]
| |
| [[File:Sequential superposition of plane waves.gif|360px|"360px"|right|thumb|The superposition of several plane waves to form a wave packet. This wave packet becomes increasingly localized with the addition of many waves. The Fourier transform is a mathematical operation that separates a wave packet into its individual plane waves. Note that the waves shown here are real for illustrative purposes only, whereas in quantum mechanics the wave function is generally complex.]]
| |
| {{multiple image
| |
| | align = right
| |
| | direction = vertical
| |
| | footer = Propagation of [[matter wave|de Broglie waves]] in 1d—real part of the [[complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the colour [[opacity (optics)|opacity]]) of finding the particle at a given point ''x'' is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the [[curvature]] reverses sign, so the amplitude begins to decrease again, and vice versa—the result is an alternating amplitude: a wave.
| |
| | image1 = Propagation of a de broglie plane wave.svg
| |
| | caption1 = [[Plane wave]]
| |
| | width1 = 250
| |
| | image2 = Propagation of a de broglie wavepacket.svg
| |
| | caption2 = [[Wave packet]]
| |
| | width2 = 250
| |
| }}
| |
| {{Main|Wave packet}}
| |
| {{Main|Schrödinger equation}}
| |
| According to the [[Matter wave|de Broglie hypothesis]], every object in the universe is a [[wave]], a situation which gives rise to this phenomenon. The position of the particle is described by a [[wave function]] <math>\Psi(x,t)</math>. The time-independent wave function of a single-moded plane wave of wavenumber ''k''<sub>0</sub> or momentum ''p''<sub>0</sub> is
| |
| | |
| :<math>\psi(x) \propto e^{ik_0 x} = e^{ip_0 x/\hbar}</math>
| |
| | |
| The [[Born rule]] states that this should be interpreted as a [[probability density function]] in the sense that the probability of finding the particle between ''a'' and ''b'' is
| |
| | |
| :<math> \operatorname P [a \leq X \leq b] = \int_a^b |\psi(x)|^2 \, \mathrm{d}x </math>.
| |
| | |
| In the case of the single-moded plane wave, <math>|\psi(x)|^2</math> is a [[uniform distribution (continuous)|uniform distribution]]. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. Consider a wave function that is a [[superposition principle|sum of many waves]], however, we may write this as
| |
| | |
| :<math>\psi(x) \propto \sum_{n} A_n e^{i p_n x/\hbar}, </math>
| |
| where ''A''<sub>''n''</sub> represents the relative contribution of the mode ''p''<sub>''n''</sub> to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized. We may take this a step further to the continuum limit, where the wave function is an [[integral]] over all possible modes
| |
| | |
| :<math>\psi(x) = \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} \phi(p) \cdot e^{i p x/\hbar}\, dp, </math>
| |
| | |
| with <math>\phi(p)</math> representing the amplitude of these modes and is called the wave function in [[momentum space]]. In mathematical terms, we say that <math>\phi(p)</math> is the ''[[Fourier transform]]'' of <math>\psi(x)</math> and that ''x'' and ''p'' are [[conjugate variables]]. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta.
| |
| | |
| One way to quantify the precision of the position and momentum is the [[standard deviation]] σ. Since <math>|\psi(x)|^2</math> is a probability density function for position, we calculate its standard deviation.
| |
| | |
| The precision of the position is improved, i.e. reduced σ<sub>x</sub>, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σ<sub>p</sub>. Another way of stating this is that σ<sub>x</sub> and σ<sub>p</sub> have an [[inverse relationship]] or are at least bounded from below. This is the uncertainty principle, the exact limit of which, is the Kennard bound. Click the ''show'' button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.
| |
| {| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
| |
| !Proof of the Kennard inequality using wave mechanics
| |
| |-
| |
| |We are interested in the [[variance]]s of position and momentum, defined as
| |
| | |
| :<math>\sigma_x^2 = \int_{-\infty}^\infty x^2 \cdot |\psi(x)|^2 \, dx - \left( \int_{-\infty}^\infty x \cdot |\psi(x)|^2 \, dx \right)^2</math>
| |
| | |
| :<math>\sigma_p^2 = \int_{-\infty}^\infty p^2 \cdot |\phi(p)|^2 \, dp - \left( \int_{-\infty}^\infty p \cdot |\phi(p)|^2 \, dp \right)^2.</math>
| |
| | |
| [[Without loss of generality]], we will assume that the [[expected value|means]] vanish, which just amounts to a shift of the origin of our coordinates. (A more general proof that does not make this assumption is given below.) This gives us the simpler form
| |
| | |
| :<math>\sigma_x^2 = \int_{-\infty}^\infty x^2 \cdot |\psi(x)|^2 \, dx</math>
| |
| :<math>\sigma_p^2 = \int_{-\infty}^\infty p^2 \cdot |\phi(p)|^2 \, dp.</math>
| |
| | |
| The function <math>f(x) = x \cdot \psi(x)</math> can be interpreted as a [[vector space|vector]] in a [[function space]]. We can define an [[inner product]] for a pair of functions ''u''(''x'') and ''v''(''x'') in this vector space:
| |
| | |
| :<math>\langle u | v \rangle = \int_{-\infty}^{\infty} u^*(x) \cdot v(x) \, dx,</math>
| |
| where the asterisk denotes the [[complex conjugate]].
| |
| | |
| With this inner product defined, we note that the variance for position can be written as
| |
| :<math>\sigma_x^2 = \int_{-\infty}^{\infty} |f(x)|^2 \, dx = \langle f | f \rangle.</math>
| |
| | |
| We can repeat this for momentum by interpreting the function <math>\tilde{g}(p)=p \cdot \phi(p)</math> as a vector, but we can also take advantage of the fact that <math>\psi(x)</math> and <math>\phi(p)</math> are Fourier transforms of each other. We evaluate the inverse Fourier transform through [[integration by parts]]:
| |
| :<math>\begin{align} g(x) &= \frac{1}{\sqrt{2 \pi \hbar}} \cdot \int_{-\infty}^{\infty} \tilde{g}(p) \cdot e^{ipx/\hbar} \, dp \\
| |
| &= \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} p \cdot \phi(p) \cdot e^{ipx/\hbar} \, dp \\
| |
| &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{\infty} \left[ p \cdot \int_{-\infty}^{\infty} \psi(x) e^{-ipx/\hbar} \, dx \right] \cdot e^{ipx/\hbar} \, dp \\
| |
| &= \frac{i}{2 \pi} \int_{-\infty}^{\infty} \left[ \cancel{ \left. \psi(x) e^{-ipx/\hbar} \right|_{-\infty}^{\infty} } - \int_{-\infty}^{\infty} \frac{d\psi(x)}{dx} e^{-ipx/\hbar} \, dx \right] \cdot e^{ipx/\hbar} \, dp \\
| |
| &= \frac{-i}{2 \pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d\psi(x)}{dx} e^{-ipx/\hbar} \, dx \, e^{ipx/\hbar} \, dp \\
| |
| &= \left( -i \hbar \frac{d}{dx} \right) \cdot \psi(x) ,\end{align}</math>
| |
| where the canceled term vanishes because the wave function vanishes at infinity. Often the term <math>-i \hbar \frac{d}{dx}</math> is called the momentum operator in position space. Applying [[Parseval's theorem]], we see that the variance for momentum can be written as
| |
| | |
| :<math>\sigma_p^2 = \int_{-\infty}^{\infty} |\tilde{g}(p)|^2 \, dp = \int_{-\infty}^{\infty} |g(x)|^2 \, dx = \langle g | g \rangle.</math>
| |
| | |
| The [[Cauchy–Schwarz inequality]] asserts that
| |
| :<math>\sigma_x^2 \sigma_p^2 = \langle f | f \rangle \cdot \langle g | g \rangle \ge |\langle f | g \rangle|^2</math>
| |
| | |
| The modulus squared of any complex number ''z'' can be expressed as
| |
| :<math>|z|^{2} = \Big(\text{Re}(z)\Big)^{2}+\Big(\text{Im}(z)\Big)^{2} \geq \Big(\text{Im}(z)\Big)^{2}=\Big(\frac{z-z^{\ast}}{2i}\Big)^{2}. </math>
| |
| we let <math>z=\langle f|g\rangle</math> and <math>z^{*}=\langle g|f\rangle</math> and substitute these into the equation above to get
| |
| :<math>|\langle f|g\rangle|^{2} \geq \bigg(\frac{\langle f|g\rangle-\langle g|f\rangle}{2i}\bigg)^{2}</math>
| |
| | |
| All that remains is to evaluate these inner products.
| |
| | |
| :<math>\begin{align}\langle f|g\rangle-\langle g|f\rangle &= \int_{-\infty}^{\infty} \psi^*(x) \, x \cdot \left(-i \hbar \frac{d}{dx}\right) \, \psi(x) \, dx \\
| |
| &{} \, \, \, \, \, - \int_{-\infty}^{\infty} \psi^*(x) \, \left(-i \hbar \frac{d}{dx}\right) \cdot x \, \psi(x) dx \\
| |
| &= i \hbar \cdot \int_{-\infty}^{\infty} \psi^*(x) \left[ \left(-x \cdot \frac{d\psi(x)}{dx}\right) + \frac{d(x \psi(x))}{dx} \right] \, dx \\
| |
| &= i \hbar \cdot \int_{-\infty}^{\infty} \psi^*(x) \left[ \left(-x \cdot \frac{d\psi(x)}{dx}\right) + \psi(x) + \left(x \cdot \frac{d\psi(x)}{dx}\right)\right] \, dx \\
| |
| &= i \hbar \cdot \int_{-\infty}^{\infty} \psi^*(x) \psi(x) \, dx \\
| |
| &= i \hbar \cdot \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx \\
| |
| &= i \hbar\end{align}</math>
| |
| | |
| Plugging this into the above inequalities, we get
| |
| :<math>\sigma_x^2 \sigma_p^2 \ge |\langle f | g \rangle|^2 \ge \left(\frac{\langle f|g\rangle-\langle g|f\rangle}{2i}\right)^2 = \left(\frac{i \hbar}{2 i}\right)^2 = \frac{\hbar^2}{4}</math>
| |
| or taking the square root
| |
| :<math>\sigma_x \sigma_p \ge \frac{\hbar}{2}.</math>
| |
| | |
| Note that the only ''physics'' involved in this proof was that <math>\psi(x)</math> and <math>\phi(p)</math> are wave functions for position and momentum, which are Fourier transforms of each other. A similar result would hold for ''any'' pair of conjugate variables.
| |
| |}
| |
| | |
| ===Matrix mechanics interpretation===
| |
| {{Main|Matrix mechanics}}
| |
| In matrix mechanics, observables such as position and momentum are represented by [[self-adjoint operator]]s. When considering pairs of observables, one of the most important quantities is the ''[[commutator]]''. For a pair of operators {{mvar|Â}} and {{mvar|B̂}}, we may define their commutator as
| |
| :<math>[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}.</math>
| |
| In the case of position and momentum, the commutator is the [[canonical commutation relation]]
| |
| :<math>[\hat{x},\hat{p}]=i \hbar.</math>
| |
| The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum [[eigenstate]]s. Let <math>|\psi\rangle</math> be a right eigenstate of position with a constant eigenvalue ''x''<sub>0</sub>. By definition, this means that <math>\hat{x}|\psi\rangle = x_0 |\psi\rangle.</math> Applying the commutator to <math>|\psi\rangle</math> yields
| |
| :<math>[\hat{x},\hat{p}] | \psi \rangle = (\hat{x}\hat{p}-\hat{p}\hat{x}) | \psi \rangle = (\hat{x} - x_0 \hat{I}) \cdot \hat{p} \, | \psi \rangle = i \hbar | \psi \rangle,</math>
| |
| where <math>\hat{I}</math> is simply the [[identity matrix|identity operator]]. Suppose for the sake of [[proof by contradiction]] that <math>|\psi\rangle</math> is also a right eigenstate of momentum, with constant eigenvalue ''p''<sub>0</sub>. If this were true, then we could write
| |
| :<math>(\hat{x} - x_0 \hat{I}) \cdot \hat{p} \, | \psi \rangle = (\hat{x} - x_0 \hat{I}) \cdot p_0 \, | \psi \rangle = (x_0 \hat{I} - x_0 \hat{I}) \cdot p_0 \, | \psi \rangle=0.</math>
| |
| On the other hand, the canonical commutation relation requires that
| |
| :<math>[\hat{x},\hat{p}] | \psi \rangle=i \hbar | \psi \rangle \ne 0.</math>
| |
| This implies that no quantum state can be simultaneously both a position and a momentum eigenstate. When a state is measured, it is projected onto an eigenstate in the basis of the observable. For example, if a particle's position is measured, then the state exists at least momentarily in a position eigenstate. This means that the state is ''not'' in a momentum eigenstate, however, but rather exists as a sum of multiple momentum basis eigenstates. In other words the momentum must be less precise. The precision may be quantified by the standard deviations, defined by
| |
| :<math>\sigma_{x}=\sqrt{\langle \hat{x}^{2} \rangle-\langle \hat{x}\rangle ^{2}}</math>
| |
| :<math>\sigma_{p}=\sqrt{\langle \hat{p}^{2} \rangle-\langle \hat{p}\rangle ^{2}}.</math>
| |
| As with the wave mechanics interpretation above, we see a tradeoff between the precisions of the two, given by the uncertainty principle.
| |
| | |
| ==Robertson–Schrödinger uncertainty relations==
| |
| The most common general form of the uncertainty principle is the ''[[Howard Percy Robertson|Robertson]] uncertainty relation''.<ref name="Robertson1929">{{Citation|last=Robertson|first=H. P.|title=The Uncertainty Principle|journal=Phys. Rev.|year=1929|volume=34|pages=163–64|bibcode = 1929PhRv...34..163R |doi = 10.1103/PhysRev.34.163 }}</ref>
| |
| | |
| For an arbitrary [[Self-adjoint operator|Hermitian operator]] <math>\hat{\mathcal{O}}</math> we can associate a standard deviation
| |
| :::<math>\sigma_{\mathcal{O}}=\sqrt{\langle \hat{\mathcal{O}}^{2} \rangle-\langle \hat{\mathcal{O}}\rangle ^{2}},</math>
| |
| where the brackets <math>\langle\mathcal{O}\rangle</math> indicate an [[expectation value (quantum mechanics)|expectation value]]. For a pair of operators {{mvar|Â}} and {{mvar|B̂}},we may define their ''[[commutator]]'' as
| |
| :::<math>[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A},</math>
| |
| In this notation, the Robertson uncertainty relation is given by
| |
| ::: <math>\sigma_{A}\sigma_{B} \geq \left| \frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle \right| = \frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle \right|.</math>
| |
| The Robertson uncertainty relation immediately [[Logical consequence|follows from]] a slightly stronger inequality, the ''Schrödinger uncertainty relation'',<ref name="Schrodinger1930">{{Citation | last = Schrödinger |first = E. | title = Zum Heisenbergschen Unschärfeprinzip | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse | volume = 14 | pages = 296–303 | year = 1930}}</ref>
| |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math>\sigma_{A}^2\sigma_{B}^2 \geq \left| \frac{1}{2}\langle\{\hat{A},\hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle \right|^{2}+ \left|\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle\right|^{2} ,</math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| where we have introduced the ''anticommutator'',
| |
| :::<math>\{\hat{A},\hat{B}\}=\hat{A}\hat{B}+\hat{B}\hat{A}.</math>
| |
| | |
| {| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
| |
| !Proof of the Schrödinger uncertainty relation
| |
| |-
| |
| |The derivation shown here incorporates and builds off of those shown in Robertson,<ref name="Robertson1929" /> Schrödinger<ref name="Schrodinger1930" /> and standard textbooks such as Griffiths.<ref name="Griffiths2005">{{Citation|last=Griffiths|first=David|title=Quantum Mechanics|year=2005|publisher=Pearson|location=New Jersey}}</ref> For any Hermitian operator <math>\hat{A}</math>, based upon the definition of [[variance]], we have
| |
| | |
| :::<math> \sigma_{A}^{2} = \langle(\hat{A}-\langle \hat{A} \rangle)\Psi|(\hat{A}-\langle \hat{A} \rangle)\Psi\rangle. </math>
| |
| | |
| we let <math>|f\rangle=|(\hat{A}-\langle \hat{A} \rangle)\Psi\rangle </math> and thus
| |
| | |
| :::<math> \sigma_{A}^{2} = \langle f|f\rangle\, .</math>
| |
| | |
| Similarly, for any other Hermitian operator <math> \hat{B} </math> in the same state
| |
| | |
| :::<math> \sigma_{B}^{2} = \langle(\hat{B}-\langle \hat{B} \rangle)\Psi|(\hat{B}-\langle \hat{B} \rangle)\Psi\rangle = \langle g|g\rangle </math>
| |
| | |
| for <math> |g\rangle=|(\hat{B}-\langle \hat{B} \rangle)\Psi \rangle.</math>
| |
| | |
| The product of the two deviations can thus be expressed as
| |
| | |
| :::{{NumBlk|:|<math> \sigma_A^2\sigma_B^2 = \langle f|f\rangle\langle g|g\rangle. </math>|{{EquationRef|1}}}}
| |
| | |
| In order to relate the two vectors <math>|f\rangle</math> and <math>|g\rangle</math>, we use the [[Cauchy–Schwarz inequality]]<ref name="Riley2006">{{Citation | last = Riley | first = K. F. | coauthors = M. P. Hobson and S. J. Bence | title = Mathematical Methods for Physics and Engineering | publisher = Cambridge | year = 2006 | pages = 246 }}</ref> which is defined as
| |
| :::<math>\langle f|f\rangle\langle g|g\rangle \geq |\langle f|g\rangle|^2, \, </math>
| |
| and thus Eq. ({{EquationNote|1}}) can be written as
| |
| :::{{NumBlk|:|<math>\sigma_A^2\sigma_B^2 \geq |\langle f|g\rangle|^2.</math>|{{EquationRef|2}}}}
| |
| Since <math> \langle f|g\rangle </math> is in general a complex number, we use the fact that the modulus squared of any complex number <math>z</math> is defined as <math>|z|^{2}=zz^{*},</math> where <math>z^{*}</math> is the complex conjugate of <math>z</math>. The modulus squared can also be expressed as
| |
| | |
| :::{{NumBlk|:|<math> |z|^2 = \Big(\text{Re}(z)\Big)^2+\Big(\text{Im}(z)\Big)^2 = \Big(\frac{z+z^\ast}{2}\Big)^2 +\Big(\frac{z-z^{\ast}}{2i}\Big)^{2}. </math>|{{EquationRef|3}}}}
| |
| | |
| we let <math>z=\langle f|g\rangle</math> and <math>z^{*}=\langle g|f\rangle</math> and substitute these into the equation above to get
| |
| :::{{NumBlk|:|<math>|\langle f|g\rangle|^{2} = \bigg(\frac{\langle f|g\rangle+\langle g|f\rangle}{2}\bigg)^{2} + \bigg(\frac{\langle f|g\rangle-\langle g|f\rangle}{2i}\bigg)^{2}</math>|{{EquationRef|4}}}}
| |
| | |
| The inner product <math>\langle f|g\rangle </math> is written out explicitly as
| |
| :::<math>\langle f|g\rangle = \langle(\hat{A}-\langle \hat{A} \rangle)\Psi|(\hat{B}-\langle \hat{B} \rangle)\Psi\rangle,</math>
| |
| and using the fact that <math>\hat{A}</math> and <math>\hat{B}</math> are Hermitian operators, we find
| |
| :<math>\langle f|g\rangle = \langle\Psi|(\hat{A}-\langle \hat{A}\rangle)(\hat{B}-\langle \hat{B}\rangle)\Psi\rangle </math>
| |
| :::<math>= \langle\Psi|(\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 </math>
| |
| :::<math>= \langle\Psi|\hat{A}\hat{B}\Psi\rangle-\langle\Psi|\hat{A}\langle \hat{B}\rangle\Psi\rangle
| |
| -\langle\Psi|\hat{B}\langle \hat{A}\rangle\Psi\rangle+\langle\Psi|\langle \hat{A}\rangle\langle \hat{B}\rangle\Psi\rangle</math>
| |
| :::<math>=\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</math>
| |
| :::<math>=\langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle\, </math>.
| |
| Similarly it can be shown that
| |
| <math>\langle g|f\rangle = \langle \hat{B}\hat{A}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle.</math>
| |
| Thus we have
| |
| :::<math>
| |
| \langle f|g\rangle-\langle g|f\rangle = \langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle-\langle \hat{B}\hat{A}\rangle+\langle \hat{A}\rangle\langle \hat{B}\rangle = \langle [\hat{A},\hat{B}]\rangle
| |
| </math>
| |
| and
| |
| :::<math>\langle f|g\rangle+\langle g|f\rangle = \langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle+\langle \hat{B}\hat{A}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle = \langle \{\hat{A},\hat{B}\}\rangle -2\langle \hat{A}\rangle\langle \hat{B}\rangle </math>.
| |
| | |
| We now substitute the above two equations above back into Eq. ({{EquationNote|4}}) and get
| |
| :::<math>
| |
| |\langle f|g\rangle|^{2}=\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle\Big)^{2}+ \Big(\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle\Big)^{2}\, .
| |
| </math>
| |
| Substituting the above into Eq. ({{EquationNote|2}}) we get the Schrödinger uncertainty relation
| |
| :::<math>
| |
| \sigma_{A}\sigma_{B} \geq \sqrt{\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle\Big)^{2}+ \Big(\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle\Big)^{2}}\,
| |
| </math>.
| |
| | |
| This proof has an issue<ref>{{Citation|last=Davidson|first=E. R.|title=On Derivations of the Uncertainty Principle|journal=J. Chem. Phys.|volume=42|year=1965|doi=10.1063/1.1696139|bibcode = 1965JChPh..42.1461D|issue=4|pages=1461 }}</ref> related to the domains of the operators involved. For the proof to make sense, the vector <math> \hat{B} |\Psi \rangle</math> has to be in the domain of the [[unbounded operator]] <math> \hat{A}</math>, which is not always the case. In fact, the Robertson uncertainty relation is false if <math>\hat{A}</math> is an angle variable and <math>\hat{B}</math> is the derivative with respect to this variable. In this example, the commutator is a nonzero constant—just as in the Heisenberg uncertainty relation—and yet there are states where the product of the uncertainties is zero.<ref name="Hall2013">{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 245 }}</ref> This issue can be overcome by using a [[variational method]] for the proof.,<ref>{{Citation|last=Jackiw|first=Roman|title=Minimum Uncertainty Product, Number‐Phase Uncertainty Product, and Coherent States|journal=J. Math. Phys.|volume=9|year=1968|doi=10.1063/1.1664585|bibcode = 1968JMP.....9..339J|issue=3|pages=339 }}</ref><ref name="CarruthersNieto">{{Citation|first=P. |last=Carruthers|last2= Nieto|first2=M. M.|title=Phase and Angle Variables in Quantum Mechanics|journal=Rev. Mod. Phys.|volume=40|year=1968|doi=10.1103/RevModPhys.40.411|bibcode = 1968RvMP...40..411C|issue=2|pages=411 }}</ref> or by working with an exponentiated version of the canonical commutation relations.<ref name="Hall2013"/>
| |
| | |
| Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operators <math>\hat{A}</math> and <math>\hat{B}</math> are [[Self-adjoint operator#Self-adjoint operators|self-adjoint operators]]. It suffices to assume that they are merely [[Self-adjoint operator#Symmetric operators|symmetric operators]]. (The distinction between these two notions is generally glossed over in the physics literature, where the term ''Hermitian'' is used for either or both classes of operators. See Chapter 9 of Hall's book<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 }}</ref> for a detailed discussion of this important but technical distinction.)
| |
| |}
| |
| | |
| 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.
| |
| | |
| *For position and linear momentum, the [[canonical commutation relation]] <math>[\hat{x},\hat{p}]=i\hbar</math> implies the Kennard inequality from above:
| |
| | |
| :: <math>\sigma_{x}\sigma_{p} \geq \frac{\hbar}{2} </math>
| |
| | |
| *For two orthogonal components of the [[angular momentum|total angular momentum]] operator of an object:
| |
| | |
| ::<math> \sigma_{J_i} \sigma_{J_j} \geq \tfrac{\hbar}{2} \left|\left\langle J_k\right\rangle\right| ~,</math>
| |
| ::where ''i'', ''j'', ''k'' are distinct and ''J''<sub>''i''</sub> denotes angular momentum along the ''x''<sub>''i''</sub> axis. This relation implies that unless all three components vanish together, only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field. Moreover, for <math> [{J_x}, {J_y}] = i \hbar \epsilon_{xyz} {J_z} </math>, a choice <math>\hat{A}= J_x, ~~\hat{B}= J_y</math>, in angular momentum multiplets, ''ψ'' = |''j'', ''m'' ⟩, bounds the [[Casimir invariant]] (angular momentum squared, <math>\langle J_x^2+ J_y^2 + J_z^2 \rangle </math>) from below and thus yields useful constraints such as ''j'' (''j'' + 1) ≥ ''m'' (''m'' + 1), and hence ''j'' ≥ ''m'', among others.
| |
| | |
| *In non-relativistic mechanics, time is privileged as an [[independent variable]]. Nevertheless, in 1945, [[Leonid Mandelshtam|L. I. Mandelshtam]] and [[Igor Tamm|I. E. Tamm]] derived a non-relativistic '''''time-energy uncertainty relation''''', as follows.<ref>[http://daarb.narod.ru/mandtamm/index-eng.html L. I. Mandelshtam, I. E. Tamm, ''The uncertainty relation between energy and time in nonrelativistic quantum mechanics'', 1945]</ref> For a quantum system in a non-stationary state {{mvar|ψ}} and an observable ''B'' represented by a self-adjoint operator <math>\hat B</math>, the following formula holds:
| |
| | |
| ::<math> \sigma_E ~ \frac{\sigma_B}{\left | \frac{\mathrm{d}\langle \hat B \rangle}{\mathrm{d}t}\right |} \ge \frac{\hbar}{2}, </math>
| |
| | |
| :where σ<sub>''E''</sub> is the standard deviation of the energy operator (Hamiltonian) in the state {{mvar|ψ}}, σ<sub>''B''</sub> stands for the standard deviation of ''B''. Although the second factor in the left-hand side has dimension of time, it is different from the time parameter that enters the [[Schrödinger equation]]. It is a ''lifetime'' of the state {{mvar|ψ}} with respect to the observable ''B'': In other words, this is the ''time interval'' (''Δt'') after which the expectation value <math>\langle\hat B\rangle</math> changes appreciably.
| |
| :An informal, heuristic meaning of the principle is the following: A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. For example, in [[Electromagnetic spectroscopy|spectroscopy]], excited states have a finite lifetime. By the time-energy uncertainty principle, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the [[Spectral linewidth|''natural linewidth'']]. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth.<ref>The broad linewidth of fast decaying states makes it difficult to accurately measure the energy of the state, and researchers have even used detuned microwave cavities to slow down the decay-rate, to get sharper peaks. {{Citation|last=Gabrielse|first =Gerald|coauthors=H. Dehmelt|title=Observation of Inhibited Spontaneous Emission|journal=Physical Review Letters|volume=55|pages=67–70|year=1985|doi=10.1103/PhysRevLett.55.67|pmid=10031682|issue=1|bibcode=1985PhRvL..55...67G}}</ref>
| |
| :The same linewidth effect also makes it difficult to specify the [[rest mass]] of unstable, fast-decaying particles in [[particle physics]]. The faster the [[particle decay]]s (the shorter its lifetime), the less certain is its mass (the larger the particle's [[Resonance (particle physics)|width]]).
| |
| | |
| *For the number of electrons in a [[superconductor]] and the [[Phase factor|phase]] of its [[Ginzburg–Landau theory|Ginzburg–Landau order parameter]]<ref>{{Citation|last=Likharev|first=K.K.|coauthors=A.B. Zorin|title=Theory of Bloch-Wave Oscillations in Small Josephson Junctions|journal=J. Low Temp. Phys.|volume=59|issue=3/4|pages=347–382|year=1985|doi=10.1007/BF00683782|bibcode=1985JLTP...59..347L}}</ref><ref>{{Citation|first=P.W.|last=Anderson|editor-last=Caianiello|editor-first=E.R.|contribution=Special Effects in Superconductivity|title=Lectures on the Many-Body Problem, Vol. 2|year=1964|place=New York|publisher=Academic Press}}</ref>
| |
| | |
| ::<math> \Delta N \Delta \phi \geq 1 ~.</math>
| |
| | |
| <!--===Energy–time uncertainty principle{{anchors|Energy-time uncertainty principle}}===
| |
| Other than the position-momentum uncertainty relation, the most important uncertainty relation is that between energy and time. The energy-time uncertainty relation is not, however, an obvious consequence of the general Robertson–Schrödinger relation. Since energy bears the same relation to time as momentum does to space in [[special relativity]], it was clear to many early founders, [[Niels Bohr]] among them, that the following relation should hold:<ref name="Heisenberg_1927"/><ref name="Heisenberg_1930"/>
| |
| | |
| ::<math> \Delta E \Delta t \gtrsim h, </math>
| |
| | |
| but it was not always obvious what <math>\Delta t </math> precisely meant. The problem is that the time at which the particle has a given state is not an operator belonging to the particle, it is a parameter describing the evolution of the system. As [[Lev Landau]] once joked "To violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!"<ref name="C Jansson">[http://arxiv.org/pdf/0802.3625 The GMc-interpretation of Quantum Mechanics], by Christian Jansson, February 25, 2008</ref>
| |
| | |
| One ''false'' formulation of the energy-time uncertainty principle says that measuring the energy of a quantum system to an accuracy <math>\Delta E</math> requires a time interval <math>\Delta t > h/\Delta E</math>. This formulation is similar to the one alluded to in Landau's joke, and was explicitly invalidated by [[Yakir Aharonov|Y. Aharonov]] and [[David Bohm|D. Bohm]] in 1961.<ref>http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Aharonov%20a.pdf</ref> The time <math>\Delta t</math> in the uncertainty relation is the time during which the system exists unperturbed, not the time during which the experimental equipment is turned on, whereas the position in the other version of the principle refers to where the particle has some probability to be and not where the observer might look.
| |
| | |
| Another common misconception is that the energy-time uncertainty principle says that the [[conservation of energy]] can be temporarily violated—energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time.<ref>Griffiths, David J. ''An Introduction to Quantum Mechanics'' Pearson / Prentice Hall (2005).</ref> Although this agrees with the ''spirit'' of [[relativistic quantum mechanics]], it is based on the false axiom that the energy of the universe is an exactly known parameter at all times. More accurately, when events transpire at shorter time intervals, there is a greater uncertainty in the energy of these events. Therefore it is not that the conservation of energy is ''violated'' when [[quantum field theory]] uses temporary electron-positron pairs in its calculations, but that the energy of quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of ''all histories'' must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.
| |
| | |
| In 1932 Dirac offered a precise definition and derivation of the time-energy uncertainty relation in a relativistic quantum theory of "events".<ref>see here, and the linked references: http://www.springerlink.com/content/nwq557633112kxk2/</ref>-->
| |
| | |
| ==Examples==
| |
| | |
| ===Quantum harmonic oscillator stationary states===
| |
| {{Main|Quantum harmonic oscillator|Stationary state}}
| |
| Consider a one-dimensional quantum harmonic oscillator (QHO). It is possible to express the position and momentum operators in terms of the [[creation and annihilation operators]]:
| |
| :<math>\hat x = \sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger})</math>
| |
| :<math>\hat p = i\sqrt{\frac{m \omega\hbar}{2}}(a^{\dagger}-a).</math>
| |
| | |
| Using the standard rules for creation and annihilation operators on the eigenstates of the QHO,
| |
| :<math>a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle</math>
| |
| :<math>a|n\rangle=\sqrt{n}|n-1\rangle,</math>
| |
| the [[variance]]s may be computed directly,
| |
| :<math>\sigma_x^2 = \frac{\hbar}{m\omega} \left( n+\frac{1}{2}\right)</math>
| |
| :<math>\sigma_p^2 = \hbar m\omega \left( n+\frac{1}{2}\right)\, .</math>
| |
| The product of these standard deviations is then
| |
| :<math>\sigma_x \sigma_p = \hbar \left(n+\frac{1}{2}\right) \ge \frac{\hbar}{2}~ .</math>
| |
| | |
| In particular, the above Kennard bound<ref name="Kennard" /> is saturated for the [[ground state]] {{math|''n''{{=}}0}}, for which the probability density is just the [[normal distribution]].
| |
| | |
| ===Quantum harmonic oscillator with Gaussian initial condition===
| |
| {{multiple image
| |
| | align = right
| |
| | direction = vertical
| |
| | footer =
| |
| | |
| Position (blue) and momentum (red) probability densities for an initially Gaussian distribution. From top to bottom, the animations show the cases Ω=ω, Ω=2ω, and Ω=ω/2. Note the tradeoff between the widths of the distributions.
| |
| | |
| | width1 = 360
| |
| | image1 = Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_balanced.gif
| |
| | width2 = 360
| |
| | image2 = Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_narrow.gif
| |
| | width3 = 360
| |
| | image3 = Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_wide.gif
| |
| }}
| |
| | |
| In a quantum harmonic oscillator of characteristic angular frequency ω, place a state that is offset from the bottom of the potential by some displacement ''x''<sub>0</sub> as
| |
| :<math>\psi(x)=\left(\frac{m \Omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \Omega (x-x_0)^2}{2\hbar}\right)},</math>
| |
| where Ω describes the width of the initial state but need not be the same as ω. Through integration over the [[Propagator#Propagator of Free Particle and Harmonic Oscillator|propagator]], we can solve for the {{Not a typo|full time}}-dependent solution. After many cancelations, the probability densities reduce to
| |
| :<math>|\Psi(x,t)|^2 \sim \mathcal{N}\left( x_0 \cos{(\omega t)} , \frac{\hbar}{2 m \Omega} \left( \cos^2{(\omega t)} + \frac{\Omega^2}{\omega^2} \sin^2{(\omega t)} \right)\right)</math>
| |
| :<math>|\Phi(p,t)|^2 \sim \mathcal{N}\left( -m x_0 \omega \sin{(\omega t)} , \frac{\hbar m \Omega}{2} \left( \cos^2{(\omega t)} + \frac{\omega^2}{\Omega^2} \sin^2{(\omega t)} \right)\right),</math>
| |
| where we have used the notation <math>\mathcal{N}(\mu, \sigma^2)</math> to denote a normal distribution of mean μ and variance σ<sup>2</sup>. Copying the variances above and applying [[list of trigonometric identities|trigonometric identities]], we can write the product of the standard deviations as
| |
| :<math>\begin{align}\sigma_x \sigma_p&=\frac{\hbar}{2}\sqrt{\left( \cos^2{(\omega t)} + \frac{\Omega^2}{\omega^2} \sin^2{(\omega t)} \right)\left( \cos^2{(\omega t)} + \frac{\omega^2}{\Omega^2} \sin^2{(\omega t)} \right)} \\
| |
| &= \frac{\hbar}{4}\sqrt{3+\frac{1}{2}\left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-\left(\frac{1}{2}\left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-1\right) \cos{(4 \omega t)}}\end{align}</math>
| |
| From the relations
| |
| :<math>\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2} \ge 2, \, \, \, |\cos{(4 \omega t)}| \le 1,</math>
| |
| we can conclude
| |
| :<math>\sigma_x \sigma_p \ge \frac{\hbar}{4}\sqrt{3+\frac{1}{2}\left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-\left(\frac{1}{2}\left(\frac{\Omega^2}{\omega^2}+\frac{\omega^2}{\Omega^2}\right)-1\right)} = \frac{\hbar}{2}.</math>
| |
| | |
| ===Coherent states===
| |
| {{Main|Coherent state}}
| |
| A coherent state is a right eigenstate of the [[annihilation operator]],
| |
| :<math>\hat{a}|\alpha\rangle=\alpha|\alpha\rangle,</math>,
| |
| which may be represented in terms of [[Fock state]]s as
| |
| :<math>|\alpha\rangle =e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle</math>
| |
| | |
| In the picture where the coherent state is a massive particle in a QHO, the position and momentum operators may be expressed in terms of the annihilation operators in the same formulas above and used to calculate the variances,
| |
| :<math>\sigma_x^2 = \frac{\hbar}{2 m \omega}</math>
| |
| :<math>\sigma_p^2 = \frac{\hbar m \omega}{2}.</math>
| |
| Therefore every coherent state saturates the Kennard bound
| |
| :<math>\sigma_x \sigma_p = \sqrt{\frac{\hbar}{2 m \omega}} \, \sqrt{\frac{\hbar m \omega}{2}} = \frac{\hbar}{2}.</math>
| |
| with position and momentum each contributing an amount <math>\sqrt{\hbar/2}</math> in a "balanced" way. Moreover every [[squeezed coherent state]] also saturates the Kennard bound although the individual contributions of position and momentum need not be balanced in general.
| |
| | |
| ===Particle in a box===
| |
| {{Main|Particle in a box}}
| |
| Consider a particle in a one-dimensional box of length <math>L</math>. The [[Particle in a box#Wavefunctions|eigenfunctions in position and momentum space]] are
| |
| :<math>\psi_n(x,t) =
| |
| \begin{cases}
| |
| A \sin(k_n x)\mathrm{e}^{-\mathrm{i}\omega_n t}, & 0 < x < L,\\
| |
| 0, & \text{otherwise,}
| |
| \end{cases}
| |
| </math>
| |
| and
| |
| :<math>\phi_n(p,t)=\sqrt{\frac{\pi L}{\hbar}}\,\,\frac{n\left(1-(-1)^ne^{-ikL}\right) e^{-i \omega_n t}}{\pi ^2 n^2-k^2 L^2},</math>
| |
| where <math>\omega_n=\frac{\pi^2 \hbar n^2}{8 L^2 m}</math> and we have used the [[de Broglie relation]] <math>p=\hbar k</math>. The variances of <math>x</math> and <math>p</math> can be calculated explicitly:
| |
| :<math>\sigma_x^2=\frac{L^2}{12}\left(1-\frac{6}{n^2\pi^2}\right)</math>
| |
| :<math>\sigma_p^2=\left(\frac{\hbar n\pi}{L}\right)^2.</math>
| |
| The product of the standard deviations is therefore
| |
| :<math>\sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}.</math>
| |
| For all <math>n=1, \, 2, \, 3\, ...</math>, the quantity <math>\sqrt{\frac{n^2\pi^2}{3}-2}</math> is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when <math>n=1</math>, in which case
| |
| :<math>\sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{\pi^2}{3}-2} \approx 0.568 \hbar > \frac{\hbar}{2}.</math>
| |
| | |
| ===Constant momentum===
| |
| {{Main|Wave packet}}
| |
| [[File:Wave function of a Gaussian state moving at constant momentum.gif|360 px|thumb|right|Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space.]]
| |
| Assume a particle initially has a [[momentum space]] wave function described by a normal distribution around some constant momentum ''p''<sub>0</sub> according to
| |
| | |
| :<math>\phi(p) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \cdot \exp{\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}\right)},</math>
| |
| | |
| where we have introduced a reference scale <math>x_0=\sqrt{\hbar/m\omega_0}</math>, with <math>\omega_0>0</math> describing the width of the distribution−−cf. [[nondimensionalization]]. If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are
| |
| | |
| :<math>\Phi(p,t) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \cdot \exp{\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}-\frac{ip^2 t}{2m\hbar}\right)},</math>
| |
| :<math>\Psi(x,t) = \left(\frac{1}{x_0 \sqrt{\pi}} \right)^{1/2} \cdot \frac{e^{-x_0^2 p_0^2 /2\hbar^2}}{\sqrt{1+i\omega_0 t}} \cdot \exp{\left(-\frac{(x-ix_0^2 p_0/\hbar)^2}{2x_0^2 (1+i\omega_0 t)}\right)}.</math>
| |
| | |
| Since <math> \langle p(t) \rangle = p_0</math> and <math>\sigma_p(t) = \hbar / x_0 \sqrt{2},</math> this can be interpreted as a particle moving along with constant momentum at arbitrarily high precision. On the other hand, the standard deviation of the position is
| |
| :<math>\sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2}</math>
| |
| such that the uncertainty product can only increase with time as
| |
| :<math>\sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2}</math>
| |
| | |
| ==Additional uncertainty relations==
| |
| | |
| ===Mixed states===
| |
| | |
| The Robertson–Schrödinger uncertainty relation may be generalized in a straightforward way to describe [[density matrix|mixed states]].<ref>{{cite web|last=Steiger|first=Nathan|title=Quantum Uncertainty and Conservation Law Restrictions on Gate Fidelity|url=http://www.physics.byu.edu/Thesis/view.aspx?id=270|publisher=Brigham Young University|accessdate=19 June 2011}}</ref>
| |
| | |
| :::<math>\sigma_{A}^{2}\sigma_{B}^{2}\geq \left(\frac{1}{2}\mathrm{tr}(\rho\{A,B\})-\operatorname{tr}(\rho A)\mathrm{tr}(\rho B)\right)^{2}+\left(\frac{1}{2i}\mathrm{tr}(\rho[A,B])\right)^{2}</math>
| |
| | |
| ===Phase space===
| |
| | |
| In the [[phase space formulation]] of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a [[Wigner quasi-probability distribution|Wigner function]] <math>W(x,p)</math> with [[Moyal product|star product]] ★ and a function ''f'', the following is generally true:<ref>{{cite doi| 10.1142/S021773230100576X|noedit}}</ref>
| |
| | |
| :<math>\langle f^* \star f \rangle =\int (f^* \star f) \, W(x,p) \, dx dp \ge 0.</math>
| |
| | |
| Choosing <math>f=a+bx+cp</math>, we arrive at
| |
| | |
| :<math>\langle f^* \star f \rangle =\begin{bmatrix}a^* & b^* & c^* \end{bmatrix}\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix}\begin{bmatrix}a \\ b \\ c\end{bmatrix} \ge 0.</math>
| |
| | |
| Since this positivity condition is true for ''all'' ''a'', ''b'', and ''c'', it follows that all the eigenvalues of the matrix are positive. The positive eigenvalues then imply a corresponding positivity condition on the [[determinant]]:
| |
| | |
| :<math>\det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix} = \det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x^2 \rangle & \left\langle xp + \frac{i\hbar}{2} \right\rangle \\ \langle p \rangle & \left\langle xp - \frac{i\hbar}{2} \right\rangle & \langle p^2 \rangle \end{bmatrix} \ge 0,</math>
| |
| | |
| or, explicitly, after algebraic manipulation,
| |
| | |
| :<math>\sigma_x^2 \sigma_p^2 = \left( \langle x^2 \rangle - \langle x \rangle^2 \right)\left( \langle p^2 \rangle - \langle p \rangle^2 \right)\ge \left( \langle xp \rangle - \langle x \rangle \langle p \rangle \right)^2 + \frac{\hbar^2}{4} ~.</math>
| |
| | |
| ===Systematic error===
| |
| The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation. Heisenberg's original version, however, was interested in ''systematic error'', incurred by a disturbance of a quantum system by the measuring apparatus, i.e., an [[observer effect]]. If we let <math>\epsilon_{\mathcal{O}}</math> represent the error (i.e., [[accuracy]]) of a measurement of an observable <math>\mathcal{O}</math> and <math>\eta_{\mathcal{O}}</math> represent its disturbance by the measurement process, then the following inequality holds:<ref name="Ozawa2003"/>
| |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math> \epsilon_A \eta_B + \epsilon_A \sigma_B + \sigma_A \eta_B \ge \left| \frac{1}{2i} \langle [\hat{A},\hat{B}] \rangle \right|</math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| In fact, Heisenberg's uncertainty principle as originally described in the 1927 formulation mentions only the first term. Applying the notation above to Heisenberg's position-momentum relation, Heisenberg's argument could be rewritten as
| |
| :::<math>\cancel{\epsilon_x \eta_p \sim \frac{\hbar}{2}} \, \, </math> (Heisenberg).
| |
| Such a formulation is both mathematically incorrect and experimentally refuted.<ref>{{Citation|last=Erhart|first=Jacqueline|coauthors=Stephan Sponar, Georg Sulyok, Gerald Badurek, Masanao Ozawa, Yuji Hasegawa|title=Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin-measurements|journal=Nature Physics|volume=8|pages=185–189|year=2012|doi=10.1038/nphys2194|arxiv = 1201.1833 |bibcode = 2012NatPh...8..185E|issue=3 }}</ref> It is also possible to derive a similar uncertainty relation combining both the statistical and systematic error components.<ref>{{Citation|last=Fujikawa|first=Kazuo|title=Universally valid Heisenberg uncertainty relation|journal=Phys. Rev. A|volume=85|year=2012|doi=10.1103/PhysRevA.85.062117|arxiv = 1205.1360 |bibcode = 2012PhRvA..85f2117F|issue=6 }}</ref>
| |
| | |
| === Entropic uncertainty principle ===
| |
| {{Main|Entropic uncertainty}}
| |
| For many distributions, the standard deviation is not a particularly natural way of quantifying the structure. For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.<ref name="CarruthersNieto" /><ref>{{Citation |first=D. |last=Judge |title=On the uncertainty relation for angle variables|journal=Il Nuovo Cimento|year=1964|volume=31|issue=2|pages=332–340|doi=10.1007/BF02733639}}</ref><ref>{{Citation |first=M. |last=Bouten|coauthors=N. Maene, P. Van Leuven|title=On an uncertainty relation for angle variables|journal=Il Nuovo Cimento|year=1965|volume=37|issue=3|pages=1119–1125|doi=10.1007/BF02773197}}</ref><ref>{{Citation |first=W. H. |last=Louisell|title=Amplitude and phase uncertainty relations|journal=Physics Letters|year=1963|volume=7|issue=1|pages=60–61|doi=10.1016/0031-9163(63)90442-6|bibcode = 1963PhL.....7...60L }}</ref> Other examples include highly [[bimodal distribution]]s, or [[unimodal distribution]]s with divergent variance.
| |
| | |
| A solution that overcomes these issues is an uncertainty based on [[entropic uncertainty]] instead of the product of variances. While formulating the [[many-worlds interpretation]] of quantum mechanics in 1957, [[Hugh Everett III]] conjectured a stronger extension of the uncertainty principle based on entropic certainty.<ref>{{Citation |last=DeWitt |first=B. S. |last2=Graham |first2=N. |year=1973 |title=The [[Many-Worlds Interpretation]] of Quantum Mechanics |location=Princeton |publisher=Princeton University Press |pages=52–53 |isbn=0-691-08126-3 }}</ref> This conjecture, also studied by Hirschman<ref>{{Citation |first=I. I., Jr. |last=Hirschman |title=A note on entropy |journal=[[American Journal of Mathematics]] |year=1957 |volume=79 |issue=1 |pages=152–156 |doi=10.2307/2372390 |postscript=. |jstor=2372390 }}</ref> and proven in 1975 by Beckner<ref name="Beckner">{{Citation |first=W. |last=Beckner |title=Inequalities in Fourier analysis |journal=[[Annals of Mathematics]] |volume=102 |issue=6 |year=1975 |pages=159–182 |doi=10.2307/1970980 |postscript=. |jstor=1970980 }}</ref> and by Iwo Bialynicki-Birula and Jerzy Mycielski<ref name="BBM">{{Citation |first=I. |last=Bialynicki-Birula|last2= Mycielski|first2=J.|title=Uncertainty Relations for Information Entropy in Wave Mechanics|journal=[[Communications in Mathematical Physics]] |volume=44 |year=1975 |pages=129 |doi=10.1007/BF01608825 |issue=2|bibcode = 1975CMaPh..44..129B }}</ref> is
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math>H_x + H_p \ge \ln (e \pi)</math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| where we have used the [[Shannon entropy]] (''not'' the quantum [[von Neumann entropy]])
| |
| | |
| :<math>H_x = - \int |\psi(x)|^2 \ln (|\psi(x)|^2 \cdot \ell ) \,dx =-\left\langle \ln (|\psi(x)|^2 \cdot \ell ) \right\rangle</math>
| |
| :<math>H_p = - \int |\phi(p)|^2 \ln (|\phi(p)|^2 \cdot \hbar / \ell ) \,dp =-\left\langle \ln (|\phi(p)|^2 \cdot \hbar / \ell ) \right\rangle</math>
| |
| | |
| for some arbitrary fixed length scale <math>\ell</math>.
| |
| | |
| From the inverse logarithmic Sobolev inequalites<ref>{{Citation |first=D. |last=Chafaï |title=Gaussian maximum of entropy and reversed log-Sobolev inequality|arxiv=math/0102227 |year=1970 |bibcode=2001math......2227C |pages=2227}}</ref>
| |
| | |
| :<math>H_x \le \frac{1}{2} \ln ( 2e\pi \sigma_x^2 / \ell^2 )~,</math>
| |
| :<math>H_p \le \frac{1}{2} \ln ( 2e\pi \sigma_p^2 \ell^2 / \hbar^2 )~,</math>
| |
| (equivalently, from the fact that normal distributions maximize the entropy of all such with a given variance),
| |
| it readily follows that this entropic uncertainty principle is ''stronger than the one based on standard deviations'', because
| |
| | |
| :<math>\sigma_x \sigma_p \ge \frac{\hbar}{2} \cdot \exp\left(H_x + H_p - \ln (e \pi) \right) \ge \frac{\hbar}{2}~.</math>
| |
| | |
| A few remarks on these inequalities. First, the choice of [[base e]] is a matter of popular convention in physics. The logarithm can alternatively be in any base, provided that it be consistent on both sides of the inequality. Second, the numerical value on the right hand side assumes the unitary convention of the Fourier transform, used throughout physics and elsewhere in this article. Third, the normal distribution saturates the inequality, and it is the only distribution with this property, because it is the [[maximum entropy probability distribution]] among those with fixed variance (cf. [[differential entropy#Maximization in the normal distribution|here]] for proof).
| |
| | |
| {| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
| |
| !Entropic uncertainty of the normal distribution
| |
| |-
| |
| |We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. The length scale can be set to whatever is convenient, so we assign
| |
| :<math>\ell=\sqrt{\frac{\hbar}{2m\omega}}</math>
| |
| :<math>\begin{align}\psi(x) &= \left(\frac{m \omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \omega x^2}{2\hbar}\right)} \\
| |
| &= \left(\frac{1}{2\pi \ell^2}\right)^{1/4} \exp{\left( -\frac{x^2}{4\ell^2}\right)} \end{align}</math>
| |
| | |
| The probability distribution is the normal distribution
| |
| :<math>|\psi(x)|^2 = \frac{1}{\ell \sqrt{2\pi}} \exp{\left( -\frac{x^2}{2\ell^2}\right)}</math>
| |
| | |
| with Shannon entropy
| |
| :<math>\begin{align}H_x &= - \int |\psi(x)|^2 \ln (|\psi(x)|^2 \cdot \ell ) \,dx \\
| |
| &= -\frac{1}{\ell \sqrt{2\pi}} \int_{-\infty}^{\infty} \exp{\left( -\frac{x^2}{2\ell^2}\right)} \ln \left[\frac{1}{\sqrt{2\pi}} \exp{\left( -\frac{x^2}{2\ell^2}\right)}\right] \, dx \\
| |
| &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \exp{\left( -\frac{u^2}{2}\right)} \left[\ln(\sqrt{2\pi}) + \frac{u^2}{2}\right] \, du\\
| |
| &= \ln(\sqrt{2\pi}) + \frac{1}{2}.\end{align}</math>
| |
| | |
| A completely analogous calculation proceeds for the momentum distribution.
| |
| :<math>\phi(p) = \left(\frac{2 \ell^2}{\pi \hbar^2}\right)^{1/4} \exp{\left( -\frac{\ell^2 p^2}{\hbar^2}\right)}</math>
| |
| :<math>|\phi(p)|^2 = \sqrt{\frac{2 \ell^2}{\pi \hbar^2}} \exp{\left( -\frac{2\ell^2 p^2}{\hbar^2}\right)}</math>
| |
| :<math>\begin{align}H_p &= - \int |\phi(p)|^2 \ln (|\phi(p)|^2 \cdot \hbar / \ell ) \,dp \\
| |
| &= -\sqrt{\frac{2 \ell^2}{\pi \hbar^2}} \int_{-\infty}^{\infty} \exp{\left( -\frac{2\ell^2 p^2}{\hbar^2}\right)} \ln \left[\sqrt{\frac{2}{\pi}} \exp{\left( -\frac{2\ell^2 p^2}{\hbar^2}\right)}\right] \, dp \\
| |
| &= \sqrt{\frac{2}{\pi}} \int_{-\infty}^{\infty} \exp{\left( -2v^2\right)} \left[\ln\left(\sqrt{\frac{\pi}{2}}\right) + 2v^2 \right] \, dv \\
| |
| &= \ln\left(\sqrt{\frac{\pi}{2}}\right) + \frac{1}{2}.\end{align}</math>
| |
| | |
| The entropic uncertainty is therefore the limiting value
| |
| :<math>\begin{align}H_x+H_p &= \ln(\sqrt{2\pi}) + \frac{1}{2} + \ln\left(\sqrt{\frac{\pi}{2}}\right) + \frac{1}{2}\\
| |
| &= 1 + \ln \pi = \ln(e\pi).\end{align}</math>
| |
| |}<!--
| |
| | |
| A measurement apparatus will have a finite resolution set by the discretization of its possible outputs into bins, with the probability of lying within one of the bins given by the Born rule. We will consider the most common experimental situation, in which the bins are of uniform size. Let ''δx'' be a measure of the spatial resolution. We take the zeroth bin to be centered near the origin, with possibly some small constant offset ''c''. The probability of lying within the jth interval of width ''δx'' is
| |
| | |
| :<math>\operatorname P[x_j]= \int_{(j-1/2)\delta x-c}^{(j+1/2)\delta x-c}|\psi(x)|^2 \, dx</math>
| |
| | |
| To account for this discretization, we can define the Shannon entropy of the wave function for a given measurement apparatus as
| |
| | |
| :<math>H_x=-\sum_{j=-\infty}^{\infty} \operatorname P[x_j] \ln \operatorname P[x_j].</math>
| |
| | |
| Under the above definition, the entropic uncertainty relation is
| |
| | |
| :<math>H_x+H_p>\ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x \delta p}{h}\right).</math>
| |
| | |
| Here we note that ''δx'' ''δp''/''h'' is a typical infinitesimal phase space volume used in the calculation of a [[partition function (statistical mechanics)|partition function]]. The inequality is also strict and not saturated. Efforts to improve this bound are an active area of research.
| |
| | |
| {| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
| |
| !Normal distribution example
| |
| |-
| |
| |We demonstrate this method first on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations.
| |
| :<math>\psi(x)=\left(\frac{m \omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \omega x^2}{2\hbar}\right)}</math>
| |
| | |
| The probability of lying within one of these bins can be expressed in terms of the [[error function]].
| |
| | |
| :<math>\begin{align}\operatorname P[x_j] &= \sqrt{\frac{m \omega}{\pi \hbar}} \int_{(j-1/2)\delta x}^{(j+1/2)\delta x} \exp{\left( -\frac{m \omega x^2}{\hbar}\right)} \, dx \\
| |
| &= \sqrt{\frac{1}{\pi}} \int_{(j-1/2)\delta x\sqrt{m \omega / \hbar}}^{(j+1/2)\delta x\sqrt{m \omega / \hbar}} e^{u^2} \, du \\
| |
| &= \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right) \right]\end{align}</math>
| |
| | |
| The momentum probabilities are completely analogous.
| |
| | |
| :<math>\operatorname P[p_j] = \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right)\delta p \cdot \frac{1}{\sqrt{\hbar m \omega}}\right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \frac{1}{\sqrt{\hbar m \omega}}\right) \right]</math>
| |
| | |
| For simplicity, we will set the resolutions to
| |
| | |
| :<math>\delta x = \sqrt{\frac{h}{m \omega}}</math>
| |
| :<math>\delta p = \sqrt{h m \omega}</math>
| |
| | |
| so that the probabilities reduce to
| |
| | |
| :<math>\operatorname P[x_j] = \operatorname P[p_j] = \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right]</math>
| |
| | |
| The Shannon entropy can be evaluated numerically.
| |
| :<math>\begin{align}H_x = H_p &= -\sum_{j=-\infty}^{\infty} \operatorname P[x_j] \ln \operatorname P[x_j] \\
| |
| &= -\sum_{j=-\infty}^{\infty} \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] \ln \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right]
| |
| &\approx 0.3226\end{align}</math>
| |
| | |
| The entropic uncertainty is indeed larger than the limiting value.
| |
| :<math>H_x+H_p \approx 0.3226 + 0.3226 = 0.6452 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.3069</math>
| |
| | |
| Note that despite being in the optimal case, the inequality is not saturated.
| |
| |}
| |
| | |
| {| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
| |
| !Sinc function example
| |
| |-
| |
| |An example of a unimodal distribution with infinite variance is the [[sinc function]]. If the wave function is the correctly normalized uniform distribution,
| |
| | |
| :<math>
| |
| \psi(x)=\begin{cases}
| |
| \frac{1}{\sqrt{2a}} & \mathrm{for}\ |x| \le a, \\[8pt]
| |
| 0 & \mathrm{for}\ |x|>a
| |
| \end{cases}
| |
| </math>
| |
| | |
| then its Fourier transform is the sinc function,
| |
| | |
| :<math>\phi(p)=\sqrt{\frac{a}{\pi \hbar}} \cdot \operatorname{sinc}\left(\frac{a p}{\hbar}\right)</math>
| |
| | |
| which yields infinite momentum variance despite having a centralized shape. The entropic uncertainty, on the other hand, is finite. Suppose for simplicity that the spatial resolution is just a two-bin measurement, ''δx'' = ''a'', and that the momentum resolution is ''δp'' = ''h''/''a''.
| |
| | |
| Partitioning the uniform spatial distribution into two equal bins is straightforward. We set the offset ''c'' = 1/2 so that the two bins span the distribution.
| |
| :<math>\operatorname P[x_0] = \int_{-a}^{0} \frac{1}{2a} \, dx = \frac{1}{2}</math>
| |
| :<math>\operatorname P[x_1] = \int_{0}^{a} \frac{1}{2a} \, dx = \frac{1}{2}</math>
| |
| :<math>H_x = -\sum_{j=0}^{1} \operatorname P[x_j] \ln \operatorname P[x_j] = -\frac{1}{2} \ln \frac{1}{2} - \frac{1}{2} \ln \frac{1}{2} = \ln 2</math>
| |
| | |
| The bins for momentum must cover the entire real line. As done with the spatial distribution, we could apply an offset. It turns out, however, that the Shannon entropy is minimized when the zeroth bin for momentum is centered at the origin. (The reader is encouraged to try adding an offset.) The probability of lying within an arbitrary momentum bin can be expressed in terms of the [[sine integral]].
| |
| | |
| :<math>\begin{align}\operatorname P[p_j] &= \frac{a}{\pi \hbar} \int_{(j-1/2)\delta p}^{(j+1/2)\delta p} \mathrm{sinc}^2\left(\frac{a p}{\hbar}\right) \, dp \\
| |
| &= \frac{1}{\pi} \int_{2\pi (j-1/2)}^{2\pi (j+1/2)} \mathrm{sinc}^2(u) \, du \\
| |
| &= \frac{1}{\pi} \left[ \operatorname {Si} ((4j+2)\pi)- \operatorname {Si} ((4j-2)\pi) \right]\end{align}</math>
| |
| | |
| The Shannon entropy can be evaluated numerically.
| |
| :<math>H_p = -\sum_{j=-\infty}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] = -\operatorname P[p_0] \ln \operatorname P[p_0]-2 \cdot \sum_{j=1}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] \approx 0.53</math>
| |
| | |
| The entropic uncertainty is indeed larger than the limiting value.
| |
| :<math>H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31</math>
| |
| |}-->
| |
| | |
| ==Harmonic analysis==
| |
| {{main|Fourier transform#Uncertainty principle}}
| |
| In the context of [[harmonic analysis]], a branch of mathematics, the uncertainty principle implies that one cannot at the same time localize the value of a function and its [[Fourier transform]]. To wit, the following inequality holds,
| |
| :<math>\left(\int_{-\infty}^\infty x^2 |f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty \xi^2 |\hat{f}(\xi)|^2\,d\xi\right)\ge \frac{\|f\|_2^4}{16\pi^2}.</math>
| |
| | |
| Further mathematical uncertainty inequalities, including the above [[entropic uncertainty]], hold between a function {{mvar|f}} and its Fourier transform {{math| ƒ̂}}.<ref>{{Citation|first1=V.|last1=Havin|first2= B.|last2=Jöricke|title=The Uncertainty Principle in Harmonic Analysis|publisher=Springer-Verlag|year=1994}}</ref><ref>{{Citation
| |
| | last1 = Folland
| |
| | first1 = Gerald
| |
| | last2 = Sitaram |first2 = Alladi
| |
| | title = The Uncertainty Principle: A Mathematical Survey
| |
| | journal = Journal of Fourier Analysis and Applications
| |
| |date=May 1997
| |
| | volume = 3
| |
| | issue = 3
| |
| | pages = 207–238
| |
| | doi = 10.1007/BF02649110
| |
| |mr=98f:42006
| |
| }}</ref><ref>{{springer|title=Uncertainty principle, mathematical|id=U/u130020|first=A|last=Sitaram|year=2001}}</ref>
| |
| | |
| ===Signal processing {{anchor|Gabor limit}}===
| |
| In the context of [[signal processing]], and in particular [[time–frequency analysis]], uncertainty principles are referred to as the '''Gabor limit''', after [[Dennis Gabor]], or sometimes the ''Heisenberg–Gabor limit''. The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be both [[time limited]] and [[band limited]] (a function and its Fourier transform cannot both have bounded domain)—see [[Time limited#Bandlimited versus timelimited|bandlimited versus timelimited]].
| |
| | |
| Stated alternatively, "One cannot simultaneously sharply localize a signal (function {{mvar|f}} ) in both the [[time domain]] and [[frequency domain]] ({{math| ƒ̂}}, its Fourier transform)".
| |
| | |
| When applied to filters, the result implies that one cannot achieve high temporal resolution and frequency resolution at the same time; a concrete example are the [[Short-time Fourier transform#Resolution issues|resolution issues of the short-time Fourier transform]]—if one uses a wide window, one achieves good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite trade-off.
| |
| | |
| Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) time–frequency plane. In practice, the Gabor limit limits the ''simultaneous'' time–frequency resolution one can achieve without interference; it is possible to achieve higher resolution, but at the cost of different components of the signal interfering with each other.
| |
| | |
| ===Benedicks's theorem===
| |
| Amrein-Berthier<ref>{{Citation
| |
| | last1 = Amrein | first1 = W.O.
| |
| | last2 = Berthier | first2 = A.M.
| |
| | year = 1977
| |
| | title = On support properties of ''L''<sup>''p''</sup>-functions and their Fourier transforms
| |
| | journal = Journal of Functional Analysis
| |
| | volume = 24 | issue = 3 | pages = 258–267
| |
| | doi = 10.1016/0022-1236(77)90056-8
| |
| | ref = harv
| |
| | postscript = .
| |
| }}</ref> and Benedicks's theorem<ref>{{Citation|first=M.|last=Benedicks|authorlink=Michael Benedicks|title=On Fourier transforms of functions supported on sets of finite Lebesgue measure|journal=J. Math. Anal. Appl.|volume=106|year=1985|issue=1|pages=180–183|doi=10.1016/0022-247X(85)90140-4}}</ref> intuitively says that the set of points where {{mvar|f}} is non-zero and the set of points where {{math| ƒ̂}} is nonzero cannot both be small.
| |
| | |
| Specifically, it is impossible for a function {{mvar|f}} in {{math|''L''<sup>2</sup>('''R''')}} and its Fourier transform {{math| ƒ̂}} to both be [[support of a function|supported]] on sets of finite [[Lebesgue measure]]. A more quantitative version is<ref>{{Citation|first=F.|last=Nazarov|authorlink=Fedor Nazarov|title=Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type,|journal=St. Petersburg Math. J.|volume=5|year=1994|pages=663–717}}</ref><ref>{{Citation|first=Ph.|last=Jaming|title=Nazarov's uncertainty principles in higher dimension|journal= J. Approx. Theory|volume=149|year=2007|issue=1|pages=30–41|doi=10.1016/j.jat.2007.04.005}}</ref>
| |
| : <math>\|f\|_{L^2(\mathbf{R}^d)}\leq Ce^{C|S||\Sigma|} \bigl(\|f\|_{L^2(S^c)} + \| \hat{f} \|_{L^2(\Sigma^c)} \bigr) ~.</math>
| |
| | |
| One expects that the factor <math>Ce^{C|S||\Sigma|}</math> may be replaced by <math>Ce^{C(|S||\Sigma|)^{1/d}}</math>,
| |
| which is only known if either {{mvar|S}} or {{mvar|Σ}} is convex.
| |
| | |
| ===Hardy's uncertainty principle===
| |
| | |
| The mathematician [[G. H. Hardy]] formulated the following uncertainty principle:<ref>{{Citation|first=G.H.|last=Hardy|authorlink=G. H. Hardy|title=A theorem concerning Fourier transforms|journal=Journal of the London Mathematical Society|volume=8|year=1933|issue=3|pages=227–231|doi=10.1112/jlms/s1-8.3.227}}</ref> it is not possible for {{mvar|f}} and {{math| ƒ̂}} to both be "very rapidly decreasing." Specifically, if {{mvar|f}} in {{math|''L''<sup>2</sup>('''R''')}} is such that
| |
| :<math>|f(x)|\leq C(1+|x|)^Ne^{-a\pi x^2}</math>
| |
| and
| |
| :<math>|\hat{f}(\xi)|\leq C(1+|\xi|)^Ne^{-b\pi \xi^2}</math> (<math>C>0,N</math> an integer),
| |
| | |
| then, if {{math|''ab'' > 1, ''f'' {{=}} 0}}, while if {{math|''ab''{{=}}1}}, then there is a polynomial {{mvar|P}} of degree {{math| ≤ ''N''}} such that
| |
| | |
| ::<math>f(x)=P(x)e^{-a\pi x^2}. \, </math>
| |
| | |
| This was later improved as follows: if {{math|''f''∈''L''<sup>2</sup>('''R'''<sup>''d''</sup>)}} is such that
| |
| | |
| :<math>\int_{\mathbf{R}^d}\int_{\mathbf{R}^d}|f(x)||\hat{f}(\xi)|\frac{e^{\pi|\langle x,\xi\rangle|}}{(1+|x|+|\xi|)^N} \, dx \, d\xi < +\infty ~,</math>
| |
| then
| |
| ::<math>f(x)=P(x)e^{-\pi\langle Ax,x\rangle} ~,</math>
| |
| where {{mvar|P}} is a polynomial of degree {{math|(''N−d'')/2}} and {{mvar|A}} is a real {{math|''d×d''}} positive definite matrix.
| |
| | |
| This result was stated in Beurling's complete works without proof and proved in Hörmander<ref>{{Citation|first=L.|last=Hörmander|authorlink=Lars Hörmander|title=A uniqueness theorem of Beurling for Fourier transform pairs|journal= Ark. Mat.|volume=29|year=1991|pages=231–240|bibcode=1991ArM....29..237H|doi=10.1007/BF02384339}}</ref> (the case <math>d=1,N=0</math>) and Bonami, Demange, and Jaming<ref>{{Citation|first1=A.|last1=Bonami|first2=B.|last2=Demange|first3=Ph.|last3=Jaming|title=Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms |journal= Rev. Mat. Iberoamericana|volume=19|year=2003|pages=23–55.|bibcode=2001math......2111B|arxiv=math/0102111| doi=10.4171/RMI/337}}</ref> for the general case. Note that Hörmander–Beurling's version implies the case {{math|''ab'' > 1}} in Hardy's Theorem while the version by Bonami–Demange–Jaming covers the full strength of Hardy's Theorem. A different proof of Beurling's theorem based on Liouville's theorem appeared in
| |
| Hedenmalm.<ref>{{Citation|first=H.|last=Hedenmalm|title=Heisenberg's uncertainty principle in the sense of Beurling|journal=J. Anal. Math.|volume=118|year=2012|pages=691–702|doi=10.1007/s11854-012-0048-9}}</ref>
| |
| | |
| A full description of the case {{math|''ab''<1}} as well as the following extension to Schwarz class distributions appears in Demange:<ref>{{Citation|first=Bruno, Demange|title=Uncertainty Principles Associated to Non-degenerate Quadratic Forms|year=2009|publisher= Société Mathématique de France|isbn=978-2-85629-297-6}}</ref>
| |
| | |
| '''Theorem.''' If a tempered distribution <math>f\in\mathcal{S}'(\R^d)</math> is such that
| |
| | |
| :<math>e^{\pi|x|^2}f\in\mathcal{S} '(\R^d)</math>
| |
| and
| |
| :<math>e^{\pi|\xi|^2}\hat f\in\mathcal{S}'(\R^d) ~,</math>
| |
| then
| |
| ::<math>f(x)=P(x)e^{-\pi\langle Ax,x\rangle} ~,</math>
| |
| for some convenient polynomial {{mvar|P}} and real positive definite matrix {{mvar|A}} of type {{math|''d × d''}}.
| |
| | |
| ==History==
| |
| | |
| [[Werner Heisenberg]] formulated the Uncertainty Principle at [[Niels Bohr]]'s institute in Copenhagen, while working on the mathematical foundations of quantum mechanics.<ref>
| |
| [http://www.aip.org/history/heisenberg/p08.htm American Physical Society online exhibit on the Uncertainty Principle]</ref>
| |
| [[File:Heisenbergbohr.jpg|thumb|Werner Heisenberg and Niels Bohr]]
| |
| In 1925, following pioneering work with [[Hendrik Kramers]], Heisenberg developed [[matrix mechanics]], which replaced the ad-hoc [[old quantum theory]] with modern quantum mechanics. The central assumption was that the classical concept of motion does not fit at the quantum level, and that [[electrons]] in an atom do not travel on sharply defined orbits. Rather, the motion is smeared out in a strange way: the [[Fourier transform]] of time only involves those frequencies that could be seen in quantum jumps.
| |
| | |
| Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact [[trajectory]], so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.
| |
| | |
| In March 1926, working in Bohr's institute, Heisenberg realized that the non-[[commutativity]] implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the [[Copenhagen interpretation]] of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a [[complementarity (physics)|complementarity]].<ref>{{Citation |first=Niels |last=Bohr |year=1958 |title=Atomic Physics and Human Knowledge |location=New York |publisher=Wiley |page=38 |isbn= |bibcode=1958AmJPh..26..596B |volume=26 |journal=American Journal of Physics |doi=10.1119/1.1934707 |issue=8 }}</ref> Any two variables that do not commute cannot be measured simultaneously—the more precisely one is known, the less precisely the other can be known. Heisenberg wrote:<blockquote>It can be expressed in its simplest form as follows: One can never know with perfect accuracy both of those two important factors which determine the movement of one of the smallest particles—its position and its velocity. It is impossible to determine accurately ''both'' the position and the direction and speed of a particle ''at the same instant''.<ref>Heisenberg, W., ''Die Physik der Atomkerne'', Taylor & Francis, 1952, p. 30.</ref></blockquote>
| |
| | |
| 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,<ref name="Heisenberg_1927"/> but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture<ref name="Heisenberg_1930">{{Citation |first=W. |last=Heisenberg |year=1930 |title={{lang|de|Physikalische Prinzipien der Quantentheorie}} |location=Leipzig |publisher=Hirzel }} English translation ''The Physical Principles of Quantum Theory''. Chicago: University of Chicago Press, 1930.</ref> he refined his principle:
| |
| | |
| :::<math>\Delta x \, \Delta p\gtrsim h\qquad\qquad\qquad (1)</math>
| |
| | |
| [[Earle Hesse Kennard|Kennard]]<ref name="Kennard" /> in 1927 first proved the modern inequality:
| |
| | |
| :::<math>\sigma_x\sigma_p\ge\frac{\hbar}{2}\quad\qquad\qquad\qquad (2)</math>
| |
| | |
| where ''ħ'' = ''h''/2π, and ''σ''<sub>''x''</sub>, ''σ''<sub>''p''</sub> are the standard deviations of position and momentum. Heisenberg only proved relation (2) for the special case of Gaussian states.<ref name="Heisenberg_1930"/>
| |
| | |
| ===Terminology and translation===
| |
| Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word, "Ungenauigkeit" ("indeterminacy"),<ref name="Heisenberg_1927" />
| |
| to describe the basic theoretical principle. Only in the endnote did he switch to the word, "Unsicherheit" ("uncertainty"). When the English-language version of Heisenberg's textbook, ''The Physical Principles of the Quantum Theory'', was published in 1930, however, the translation "uncertainty" was used, and it became the more commonly used term in the English language thereafter.<ref>{{Citation |first=David |last=Cassidy |year=2009 |title=Beyond Uncertainty: Heisenberg, Quantum Physics, and the Bomb |location= New York |publisher=Bellevue Literary Press |page=185 |isbn= |bibcode=2010PhT....63a..49C |last2=Saperstein |first2=Alvin M. |volume=63 |journal=Physics Today |doi=10.1063/1.3293416 }}</ref>
| |
| | |
| ===Heisenberg's microscope===
| |
| [[Image:Heisenberg gamma ray microscope.svg|thumb|200px|right|Heisenberg's gamma-ray microscope for locating an electron (shown in blue). The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle ''θ''. The scattered gamma-ray is shown in red. Classical [[optics]] shows that the electron position can be resolved only up to an uncertainty Δ''x'' that depends on ''θ'' and the wavelength ''λ'' of the incoming light.]]
| |
| {{Main|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 an imaginary microscope as a measuring device.<ref name="Heisenberg_1930"/>
| |
| | |
| He imagines an experimenter trying to measure the position and momentum of an [[electron]] by shooting a [[photon]] at it.
| |
| :Problem 1 – If the photon has a short [[wavelength]], and therefore, a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long [[wavelength]] and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
| |
| | |
| :Problem 2 – If a large [[aperture]] is used for the microscope, the electron's location can be well resolved (see [[Rayleigh criterion]]); but by the principle of [[conservation of momentum]], the transverse momentum of the incoming photon 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 imply 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 [[Planck's constant]].<ref>{{Citation |last=Tipler |first=Paul A. |first2=Ralph A. |last2=Llewellyn |title=Modern Physics |edition=3rd |publisher=W. H. Freeman and Co. |year=1999 |isbn=1-57259-164-1 |chapter=5–5 }}</ref> Heisenberg did not care to formulate the uncertainty principle as an exact limit (which is elaborated below), 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.
| |
| | |
| ==Critical reactions==
| |
| {{Main|Bohr–Einstein debates}}
| |
| | |
| The Copenhagen interpretation of quantum mechanics and Heisenberg's Uncertainty Principle were, in fact, seen as twin targets by detractors who believed in an underlying [[determinism]] and [[Scientific realism|realism]]. According to the [[Copenhagen interpretation]] of quantum mechanics, there is no fundamental reality that the [[quantum state]] describes, just a prescription for calculating experimental results. There is no way to say what the state of a system fundamentally is, only what the result of observations might be.
| |
| | |
| [[Albert Einstein]] believed that randomness is a reflection of our ignorance of some fundamental property of reality, while [[Niels Bohr]] believed that the probability distributions are fundamental and irreducible, and depend on which measurements we choose to perform. [[Bohr–Einstein debates|Einstein and Bohr debated]] the uncertainty principle for many years. Some experiments within the first decade of the twenty-first century have cast doubt on how extensively the uncertainty principle applies.<ref>R&D Magazine & University of Toronto, September 10, 2012 [http://www.rdmag.com/News/2012/09/General-Science-Scientists-Cast-Doubt-On-The-Uncertainty-Principle/?et_cid=2840724&et_rid=282345818&linkid=http%3A%2F%2Fwww.rdmag.com%2FNews%2F2012%2F09%2FGeneral-Science-Scientists-Cast-Doubt-On-The-Uncertainty-Principle%2F Scientists cast doubt on the uncertainty principle] retrieved Sept 10, 2012</ref>
| |
| | |
| ===Einstein's slit===
| |
| The first of Einstein's [[thought experiment]]s challenging the uncertainty principle went as follows:
| |
| | |
| :Consider a particle passing through a slit of width {{mvar|d}}. The slit introduces an uncertainty in momentum of approximately {{math|''h''/''d''}} because the particle passes through the wall. But let us determine the momentum of the particle by measuring the recoil of the wall. In doing so, we find the momentum of the particle to arbitrary accuracy by conservation of momentum.
| |
| | |
| Bohr's response was that the wall is quantum mechanical as well, and that to measure the recoil to accuracy {{math|Δ''p''}}, the momentum of the wall must be known to this accuracy before the particle passes through. This introduces an uncertainty in the position of the wall and therefore the position of the slit equal to {{math|''h''/Δ''p''}}, and if the wall's momentum is known precisely enough to measure the recoil, the slit's position is uncertain enough to disallow a position measurement.
| |
| | |
| A similar analysis with particles diffracting through multiple slits is given by [[Richard Feynman]].<ref>Feynman lectures on Physics, vol 3, 2–2</ref>
| |
| | |
| In another thought experiment Lawrence Marq Goldberg theorized that one could, for example, determine the position of a particle and then travel back in time to a point before the first reading to measure the velocity, then time travel back to a point before the second (earlier) reading was taken to deliver the resulting measurements before the particle was disturbed so that the measurements did not need to be taken. This, of course, would result in a temporal paradox. But it does support his contention that "the problems inherent to the uncertainly principle lay in the measuring not in the "uncertainty" of physics."
| |
| | |
| ===Einstein's box===
| |
| Bohr was present when Einstein proposed the thought experiment which has become known as [[Einstein's box]]. Einstein argued that "Heisenberg's uncertainty equation implied that the uncertainty in time was related to the uncertainty in energy, the product of the two being related to [[Planck's constant]]."<ref name="Gamow">Gamow, G., ''The great physicists from Galileo to Einstein'', Courier Dover, 1988, p.260.</ref> Consider, he said, an ideal box, lined with mirrors so that it can contain light indefinitely. The box could be weighed before a clockwork mechanism opened an ideal shutter at a chosen instant to allow one single photon to escape. "We now know, explained Einstein, precisely the time at which the photon left the box."<ref>Kumar, M., ''Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality'', Icon, 2009, p. 282.</ref> "Now, weigh the box again. The change of mass tells the energy of the emitted light. In this manner, said Einstein, one could measure the energy emitted and the time it was released with any desired precision, in contradiction to the uncertainty principle."<ref name="Gamow" />
| |
| | |
| Bohr spent a sleepless night considering this argument, and eventually realized that it was flawed. He pointed out that if the box were to be weighed, say by a spring and a pointer on a scale, "since the box must move vertically with a change in its weight, there will be uncertainty in its vertical velocity and therefore an uncertainty in its height above the table. ... Furthermore, the uncertainty about the elevation above the earth's surface will result in an uncertainty in the rate of the clock,"<ref>Gamow, G., ''The great physicists from Galileo to Einstein'', Courier Dover, 1988, p. 260–261.</ref> because of Einstein's own theory of [[Gravitational time dilation|gravity's effect on time]].
| |
| "Through this chain of uncertainties, Bohr showed that Einstein's light box experiment could not simultaneously measure exactly both the energy of the photon and the time of its escape."<ref>Kumar, M., ''Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality'', Icon, 2009, p. 287.</ref>
| |
| | |
| ===EPR paradox for entangled particles===
| |
| Bohr was compelled to modify his understanding of the uncertainty principle after another thought experiment by Einstein. In 1935, Einstein, Podolsky and Rosen (see [[EPR paradox]]) published an analysis of widely separated [[Quantum entanglement|entangled]] particles. Measuring one particle, Einstein realized, would alter the probability distribution of the other, yet here the other particle could not possibly be disturbed. This example led Bohr to revise his understanding of the principle, concluding that the uncertainty was not caused by a direct interaction.<ref>{{Citation |first=Walter |last=Isaacson |year=2007 |title=Einstein: His Life and Universe |location=New York |publisher=Simon & Schuster |page=452 |isbn=978-0-7432-6473-0 }}</ref>
| |
| | |
| But Einstein came to much more far-reaching conclusions from the same thought experiment. He believed the "natural basic assumption" that a complete description of reality, would have to predict the results of experiments from "locally changing deterministic quantities", and therefore, would have to include more information than the maximum possible allowed by the uncertainty principle.
| |
| | |
| In 1964, [[John Stewart Bell|John Bell]] showed that this assumption can be falsified, since it would imply a certain inequality between the probabilities of different experiments. Experimental results confirm the predictions of quantum mechanics, ruling out Einstein's basic assumption that led him to the suggestion of his ''hidden variables''. Ironically this fact is one of the best pieces of evidence supporting [[Karl Popper]]'s philosophy of [[Falsifiability|invalidation of a theory by falsification-experiments]]. That is to say, here Einstein's "basic assumption" became falsified by [[Bell test experiments|experiments based on Bell's inequalities]]. For the objections of Karl Popper against the Heisenberg inequality itself, see below.
| |
| | |
| While it is possible to assume that quantum mechanical predictions are due to nonlocal, hidden variables, and in fact [[David Bohm]] invented such a formulation, this resolution is not satisfactory to the vast majority of physicists. The question of whether a random outcome is predetermined by a nonlocal theory can be philosophical, and it can be potentially intractable. If the hidden variables are not constrained, they could just be a list of random digits that are used to produce the measurement outcomes. To make it sensible, the assumption of nonlocal hidden variables is sometimes augmented by a second assumption—that the [[size of the observable universe]] puts a limit on the computations that these variables can do. A nonlocal theory of this sort predicts that a [[quantum computer]] would encounter fundamental obstacles when attempting to factor numbers of approximately 10,000 digits or more; a potentially [[Shor's algorithm|achievable task]] in quantum mechanics.<ref>[[Gerardus 't Hooft]] has at times advocated this point of view.</ref>
| |
| | |
| === Popper's criticism ===
| |
| {{main|Popper's experiment}}
| |
| [[Karl Popper]] approached the problem of indeterminacy as a logician and [[Philosophical realism|metaphysical realist]].<ref name="Popper1959">{{Citation | last1 = Popper | first1 = Karl | authorlink1 = Karl Popper | title = [[The Logic of Scientific Discovery]] | publisher = Hutchinson & Co. | year = 1959}}</ref> He disagreed with the application of the uncertainty relations to individual particles rather than to [[Quantum ensemble|ensembles]] of identically prepared particles, referring to them as "statistical scatter relations".<ref name="Popper1959" /><ref name="Jarvie2006">{{Citation | last1 = Jarvie | first1 = Ian Charles | last2 = Milford | first2 = Karl | last3 = Miller | first3 = David W | title = Karl Popper: a centenary assessment, | volume = 3 | publisher = Ashgate Publishing | year = 2006 | isbn = 978-0-7546-5712-5}}</ref> In this statistical interpretation, a ''particular'' measurement may be made to arbitrary precision without invalidating the quantum theory. This directly contrasts with the [[Copenhagen interpretation]] of quantum mechanics, which is [[Determinism|non-deterministic]] but lacks local hidden variables.
| |
| | |
| In 1934, Popper published ''Zur Kritik der Ungenauigkeitsrelationen'' (''Critique of the Uncertainty Relations'') in ''[[Naturwissenschaften]]'',<ref name="Popper1934">{{Citation | title = Zur Kritik der Ungenauigkeitsrelationen (Critique of the Uncertainty Relations) | journal = Naturwissenschaften | year = 1934 | first = Karl | last = Popper | coauthors = Carl Friedrich von Weizsäcker | volume = 22 | issue = 48 | pages = 807–808 | doi=10.1007/BF01496543|bibcode = 1934NW.....22..807P | postscript = . }}</ref> and in the same year ''[[The Logic of Scientific Discovery|Logik der Forschung]]'' (translated and updated by the author as ''The Logic of Scientific Discovery'' in 1959), outlining his arguments for the statistical interpretation. In 1982, he further developed his theory in ''Quantum theory and the schism in Physics'', writing:
| |
| <blockquote>[Heisenberg's] formulae are, beyond all doubt, derivable ''statistical formulae'' of the quantum theory. But they have been ''habitually misinterpreted'' by those quantum theorists who said that these formulae can be interpreted as determining some upper limit to the ''precision of our measurements''.[original emphasis]<ref>Popper, K. ''Quantum theory and the schism in Physics'', Unwin Hyman Ltd, 1982, pp. 53–54.</ref></blockquote>
| |
| | |
| Popper proposed an experiment to [[Falsifiability|falsify]] the uncertainty relations, although he later withdrew his initial version after discussions with [[Carl Friedrich von Weizsäcker|Weizsäcker]], [[Werner Heisenberg|Heisenberg]], and [[Albert Einstein|Einstein]]; this experiment may have influenced the formulation of the [[EPR paradox|EPR experiment]].<ref name="Popper1959" /><ref name="Mehra2001">{{Citation | last1 = Mehra | first1 = Jagdish | last2 = Rechenberg | first2 = Helmut | title = The Historical Development of Quantum Theory | publisher = Springer | year = 2001 | isbn = 978-0-387-95086-0}}</ref>
| |
| | |
| === Many-worlds uncertainty ===
| |
| {{main|Many-worlds interpretation}}
| |
| | |
| The [[many-worlds interpretation]] originally outlined by [[Hugh Everett III]] in 1957 is partly meant to reconcile the differences between the Einstein and Bohr's views by replacing Bohr's [[wave function collapse]] with an ensemble of deterministic and independent universes whose ''distribution'' is governed by [[wave function]]s and the [[Schrödinger equation]]. Thus, uncertainty in the many-worlds interpretation follows from each observer within any universe having no knowledge of what goes on in the other universes.
| |
| | |
| === Free will ===
| |
| {{main| Two-stage model of free will}}
| |
| | |
| Some scientists including [[Arthur Compton]]<ref>A. H. Compton, ''Science'', 74, 1911, August 14, 1931</ref> and [[Martin Heisenberg]]<ref>Martin Heisenberg, "Is Free Will an Illusion?", ''Nature'', 459, (May 2009): 164–165</ref> have suggested that the uncertainty principle, or at least the general probabilistic nature of quantum mechanics, could be evidence for the two-stage model of free will. The standard view, however, is that apart from the basic role of quantum mechanics as a foundation for chemistry, [[Quantum biology|nontrivial biological mechanisms requiring quantum mechanics]] are unlikely, due to the rapid decoherence time of quantum systems at room temperature.<ref>P.C.W. Davies, "Does quantum mechanics play a non-trivial role in life?", ''BioSystems'', 78, (2004): 69–79</ref>
| |
| | |
| ==See also==
| |
| {{div col}}
| |
| * [[Canonical commutation relation]]
| |
| * [[Correspondence principle]]
| |
| * [[Correspondence rules]]
| |
| * [[Gromov's non-squeezing theorem]]
| |
| * [[Finite fourier transform#Uncertainty principle]]
| |
| * [[Heisenbug]]
| |
| * [[Introduction to quantum mechanics]]
| |
| * [[Operationalization]]
| |
| * [[Observer effect (information technology)]]
| |
| * [[Observer effect (physics)]]
| |
| * [[Quantum indeterminacy]]
| |
| * [[Quantum tunnelling]]
| |
| * ''[[The Part and The Whole]]'' (book)
| |
| * [[Weak measurement]]
| |
| {{div col end}}
| |
| | |
| ==Notes==
| |
| {{Reflist|colwidth=30em}}
| |
| | |
| ==External links==
| |
| *{{springer|title=Uncertainty principle|id=p/u095100}}
| |
| *[http://www.lightandmatter.com/html_books/6mr/ch04/ch04.html Matter as a Wave] – a chapter from an online textbook
| |
| *[http://arxiv.org/abs/quant-ph/0609163 Quantum mechanics: Myths and facts]
| |
| *[http://plato.stanford.edu/entries/qt-uncertainty/ Stanford Encyclopedia of Philosophy entry]
| |
| *[http://www.mathpages.com/home/kmath488/kmath488.htm Fourier Transforms and Uncertainty] at MathPages
| |
| *[http://www.aip.org/history/heisenberg/p08.htm aip.org: Quantum mechanics 1925–1927 – The uncertainty principle]
| |
| *[http://scienceworld.wolfram.com/physics/UncertaintyPrinciple.html Eric Weisstein's World of Physics – Uncertainty principle]
| |
| *[http://math.ucr.edu/home/baez/uncertainty.html John Baez on the time-energy uncertainty relation]
| |
| *[http://daarb.narod.ru/tcpr-eng.html The certainty principle]
| |
| *[http://www.scientificamerican.com/article.cfm?id=common-interpretation-of-heisenbergs-uncertainty-principle-is-proven-false Common Interpretation of Heisenberg's Uncertainty Principle Is Proved False]
| |
| | |
| {{Positivism}}
| |
| | |
| {{DEFAULTSORT:Uncertainty Principle}}
| |
| [[Category:Concepts in physics]]
| |
| [[Category:Quantum mechanics]]
| |
| [[Category:Principles]]
| |
| [[Category:Mathematical physics]]
| |
| | |
| {{Link GA|zh}}
| |