# Uncertainty principle

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.

The uncertainty principle, also known as Heisenberg's uncertainty principle in quantum mechanics,, 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 x and momentum p, can be known simultaneously.

Historically, the uncertainty principle has been confused with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system. Heisenberg utilized such an observer effect at the quantum level as a physical "explanation" of quantum uncertainty. It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wave-like systems, and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. This article contains some historical observer-effect quotations, as well as some related and early pre-uncertainty era quotations.

## Quotes

• If quantum mechanics is right, there is no way to get around the uncertainty principle. The reason that the electron’s probability wave spread so much after we confined it, Heisenberg would argue, is that its momentum became almost completely indeterminate. In a manner of speaking, it headed off in all directions.
• At the heart of the quantum revolution is Heisenberg's uncertainty principle... roughly... all physical quantities... are subject to unpredictable fluctuations, so that their values are not precisely defined. ...[e.g., we are] free to measure [position x and the momentum p of a quantum particle] to arbitrary precision, but they cannot possess precise values simultaneously. The spread, or uncertainty, in their values, denoted by Ax and Ap... are such that [their] product... cannot be less than... Planck's constant [after Max Planck], numerically very small... so that quantum effects are generally only important in the atomic domain... not... in daily life. ...[T]his uncertainty is inherent in nature and not merely the result of technological limitations in measurement.
• Paul Davies, (1989) Introduction to Physics and Philosophy (1958)
• The uncertainty has deep implications. For example, it means that a quantum particle does not move along a well-defined path through space. ...The smearing of position and momentum leads to an inherent indeterminism in the behaviour of quantum systems. ...[T]he experimenter may fire an electron at a target and find that it scatters to the left, then, on repeating... the next electron scatters to the right. Quantum mechanics still enables the relative probabilities of the alternatives to be specified precisely. ...It can make definite predictions about ensembles of identical systems, but it can generally tell us nothing definite about an individual system.
• Paul Davies, (1989) Introduction to Physics and Philosophy (1958)
• Relativity principles require us to associate mass with the energy of radiation, and it is reasonable to suppose... an exchange of momentum... [T]he exchange of momentum between free electrons and radiation is very similar to the exchange... when two particles collide. ...[A] beam of light should be considered as an assembly of "units", each or which [using light frequency (${\displaystyle \nu }$), Planck's constant (${\displaystyle h}$), speed of light (${\displaystyle c}$)] possesses energy (${\displaystyle W}$), momentum (${\displaystyle p}$), and mass (${\displaystyle m}$), given by
${\displaystyle W=h\nu ;\;p={\frac {h\nu }{c}};\;m={\frac {h\nu }{c^{2}}}.\quad }$ 17(14)...
This general picture was first suggested by Einstein... The units are now called photons... [T]he spreading of light by diffraction cannot be permanently concentrated in a small volume like the energy of a material particle. ...The pressure ${\displaystyle p}$, exerted by a parallel beam incident normally on a body which completely absorbs it, is...
${\displaystyle p=\rho _{p},\quad }$ ...17(15)
where ${\displaystyle \rho _{p}}$ is the energy per unit volume of the incident radiation. ...[C]onsider the radiation pressure of a parallel beam of light, incident on an absorbing body... the light is of frequency ${\displaystyle \nu }$ and... there are ${\displaystyle N}$ quanta per unit volume. Then...
${\displaystyle \rho _{p}=Nh\nu .\quad }$ ...17(18)
[A]ll the quanta in a cylinder of volume ${\displaystyle c}$ [speed of light multiplied by unit area] cubic centimetres are incident upon unit area of the surface in one second, the pressure...
${\displaystyle p=NcP,\quad }$ ...17(19)
where ${\displaystyle P}$ is the momentum of one photon. Combining...
${\displaystyle P={\frac {h\nu }{c}}={\frac {h}{\lambda }}}$.
[Experimental] results... for isotropic radiation are in agreement...
• R. W. Ditchburn, Light (1953) pp. 553-559.
• Suppose... motion of an electron in the absence of a field of force, is to be investigated... by testing the validity of [no force implies zero acceleration]...
${\displaystyle {\frac {d^{2}q}{dt^{2}}}=0,\quad }$ ...18(3)
...${\displaystyle q}$ ...the position of the particle at time ${\displaystyle t}$. The... procedure is to measure the position and momentum of the electron at... time ${\displaystyle t=t_{0}}$... to obtain two "initial conditions" which can be inserted in the solution of 18(3)... then calculate the position and momentum at some later time... and see if the calculation agrees with... observation... Suppose we observe... with light of wavelength ${\displaystyle \lambda }$. ...[D]iffraction of the wave sets the limit to the accuracy of a position measurement...
${\displaystyle \vartriangle q\sim {\frac {\lambda }{2sin\theta }},\quad }$ ...18(4)
where ${\displaystyle \vartriangle q}$ is the probable error in... ${\displaystyle q}$, and ${\displaystyle \theta }$ is the semi-angle of the cone of rays accepted by the microscope... [and] ${\displaystyle \sim }$ means "at least of the order of magnitude of". The experiment of Compton... shows that the interaction... involves an exchange of momentum. We may assume that the momenta... were exactly known before their interaction, but... [those] after the interaction depends on the accuracy [of the] momentum exchanged during the interaction. [T]he photon enters the microscope, and... we know its direction... within an angle ${\displaystyle 2\theta }$. Any attempt [to reduce] the effective aperture... increases ${\displaystyle \vartriangle q}$. Thus... the momentum of the photon in the plane [in which q is measured] perpendicular to the axis of the microscope... is uncertain by an amount
${\displaystyle \vartriangle p\sim {\frac {2h\nu }{c}}sin\theta \quad }$. ...18(5)
The momentum of the particle after the interaction is uncertain by ${\displaystyle \vartriangle p}$. Combining... we have
${\displaystyle \vartriangle p\vartriangle q\sim {\frac {\lambda }{2sin\theta }}{\frac {2h\nu }{c}}sin\theta }$,
i.e.,
${\displaystyle \vartriangle p\vartriangle q\sim h\quad }$. ...18(6)
• R. W. Ditchburn, Light (1953) pp. 584-585.
• Une loi de Physique possède une certitude beaucoup moins immédiate et beaucoup plus difficile à apprécier qu'une loi de sens commun; mais elle surpasse cette dernière par la précision minutieuse et détaillée de ses prédictions. ...Cette minutie dans le détail, les lois de la Physique ne la peuvent acquérir qu'en sacrifiant quelque chose de la certitude fixe et absolue des lois de sens commun. Entre la précision et la certitude il y a une sorte de compensation; l'une ne peut croître qu'au détriment de l'autre.
A law of physics possesses a certainty much less immediate and much more difficult to estimate than a law of common sense, but it surpasses the latter by the minute and detailed precision of its predictions. ...The laws of physics can acquire this minuteness of detail only by sacrificing something of the fixed and absolute certainty of common-sense laws. There is a sort of balance between precision and certainty; one cannot be increased except to the detriment of the other.
• Pierre Duhem, La Théorie Physique: son Objet, et sa Structure (1906) p. 292. [Duhem's Law]
Tr. Philip P. Wiener, The Aim and Structure of Physical Theory (1954)
• In September 1973, while I was visiting Moscow, I discussed black holes with two leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. They convince me that, according to the uncertainty principle, rotating black holes should create and emit particles. I believed their arguments on physical grounds but did not like the mathematical way in which they calculated the emission. I therefore set about devising a better mathematical treatment... when I did the calculation I found to my surprise and annoyance, that even nonrotating black holes should create and emit particles at a steady rate. ...[What finally convinced me that the emission was real was that the spectrum of the emitted particles was exactly that which would be emitted by a hot body... at exactly... the correct rate to prevent violations of the second law.
• [T]he laws of physics need not break down at the origin of the universe. The state of the universe and its contents, like ourselves, are completely determined by the laws of physics, up to the limit set by the uncertainty principle. So much for free will.
• Stephen Hawking, The Cambridge Lectures (1996) p. 94.
• Ultimately... one would hope to find a complete, consistent, unified theory that would include all... partial theories as approximations... "the unification of physics." Einstein spent most of his later years unsuccessfully searching... Einstein refused to believe in the reality of quantum mechanics, despite the important role he played in its development. Yet it seems that the uncertainty principle is a fundamental feature of the universe... A successful unified theory must... incorporate this principle.
• It must have been one evening after midnight when I suddenly remembered my conversation with Einstein and particularly his statement, "It is the theory which decides what we can observe." I was immediately convinced that the key to the gate that had been closed for so long must be sought right here. I decided to go on a nocturnal walk through Faelled Park and to think further about the matter. We had always said so glibly that the path of the electron in the cloud chamber could be observed. But perhaps what we really observed was something much less. Perhaps we merely saw a series of discrete and ill-defined spots through which the electron had passed. In fact, all we do see in the cloud chamber are individual water droplets which must certainly be much larger than the electron. The right question should therefore be: Can quantum mechanics represent the fact that an electron finds itself approximately in a given place and that it moves approximately with a given velocity, and can we make these approximations so close that they do not cause experimental difficulties?
• Instead of Newtonian certainty and determinism, quantum theory answers our questions with probability and statistics. Classical physics told us precisely where Mars was to be found. Quantum theory sends us to the gambling table to locate an electron in an atom. Then there's Heisenberg's uncertainty principle, which places an ultimate limit on our knowledge of the microworld and tells us that we can make no measurment without affecting the result.
• Roger S. Jones, Physics for the Rest of Us (1992) p. 155.
• Although Heisenberg began thinking about an electron track in a cloud chamber, he came to realize that the problem of locating an electron is not one of instrumentation... [H]e found that it was never possible... to measure the precise path of an electron: for example, the droplet size is too large in a cloud chamber. Or if you try to "see" and electron with light (or x-rays), then the light photons strike the electron like billiard balls and randomly change the electron's position. ...It was ...a limit in principle on the accuracy ...a theoretical limit—an irreducible degree of uncertainty or Indeterminacy—in measuring the simultaneous position and motion of an electron. ...[T]he two measurements—of position and motion—counterbalance one another. The better you measure the position, the worse you will be able to measure the motion, and vice versa. If you could measure the position (or motion) perfectly, you would know nothing about the motion (position).
The uncertainty principle thus seems to reconcile the particle-wave duality. The better you know the position... the more localized... the more it seems to act like a particle. Alternatively, if you know the motion or speed very well... [position] is diffuse, like a wave. ...[A] different point of view on duality but still ...a paradoxical fact of life.
• Roger S. Jones, Physics for the Rest of Us (1992) pp. 160-161.
• Heisenberg considered the observation through a microscope of an electron struck by light of an appropriate frequency. The position of the electron could be determined more precisely by increasing the frequency of the light, but the higher the frequency, the larger the jolt to the electron, and, hence, the greater the indeterminacy in the measurement of the electron's velocity. Conversely, the velocity could be determined more precisely by using light of a lower frequency, but the lower the frequency, the greater the indeterminacy in the measurement of the position. On the basis of these considerations, Heisenbery boldly affirmed the acausality of quantum mechanics: For to predict the future, you had to know everything about the present, and according to quantum mechanics, Heisenberg asserted, "We cannot, as a matter of principle, know the present in all its details."
• Daniel Kevles, The Physicists: The History of a Scientific Community in Modern America (1971) p. 166. Ref: Heisenberg quote, see also, Physics and Beyond (1971) pp. 72-78, Tr. Arnold J. Pomerans.
• Werner Heisenberg not only discovered but proved that in certain subatomic situations neither classical objectivity nor mechanical causality applied: that the act of the physicists observation (more exactly: his attempts at measurement) interfered either with the movement or with the situation of the object—which meant, among other things, a big crack in the fundament of Descartes's and Newton's objectivism and determinism. In other words: the study of the "reality" of matter was inseparable from the interference (and from the mind and purpose) of the scientist.
• Une hypothèse qui permet de prévoir certains effets qui se reproduisent toujours ressemble absolument à une vérité démontrée. Le système de Newton ne repose guère sur un autre fondement. Si en réalité et de l aveu du.
[A hypothesis which permits the prediction of certain effects that always reoccur under certain conditions does, in its way amount to a demonstrable certainty. Even the Newtonian system had no more than such a foundation.]
• Alexis de Tocqueville, Letter to de Gobineau (August 5, 1858) Correspondance entre Alexis de Tocqueville et Arthur de Gobineau 1843-1859 (1908) p. 330.
[Tr. John Lukacs, At the End of an Age (2002)]