Uncertainty principle
subject: Category:Quantum mechanics
image: Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_narrow.gif
description: fundamentale Relation der Quantenmechanik @de
caption: Wave packet
description: Werner Heisenbergin vuonna 1927 esittämä kvanttimekaniikan perusperiaate @fi
image: Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_balanced.gif
description: 量子力学における基礎的原理 @ja
footer: , and
title: Uncertainty principle
footer: Propagation of de Broglie waves in 1d—real part of the complex amplitude is blue, imaginary part is green. The probability 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.
label: Prinsip ketidakpastian Heisenberg @in
footer: ,
label: Princip neurčitosti @cs
description: nguyên lý cơ học lượng tử @vi
image: Propagation of a de broglie wavepacket.svg
indent: :
footer: Ω = 2ω
description: mätosäkerhet inom kvantfysiken @sv
label: Onzekerheidsrelatie van Heisenberg @nl
description: princípio fundamental na física quântica @pt
direction: vertical
border colour: #0073CF
label: مبدأ الريبة @ar
width: 360
cellpadding: 6
background colour: #F5FFFA
subject: Category:1927 in Germany
subject: Category:1927 in science
reason: What printing/edition does this page number refer to? Use year for that, and orig-year for original publication date
subject: Category:Werner Heisenberg
label: Αρχή της απροσδιοριστίας @el
label: Principi d'incertesa de Heisenberg @ca
description: 양자역학에서, 비가환인 두 관측가능량을 한꺼번에 정확히 측정할 수 없다는 원리 @ko
description: কোয়ান্টাম বলবিজ্ঞানের এক মৌলিক নীতি @bn
subject: Category:Inequalities
label: Principio di indeterminazione di Heisenberg @it
description: osnovno načelo v kvantni fiziki @sl
date: December 2024
description: عدم قطعیت در تعیین سرعت یا مکان ذره @fa
label: 不確定性原理 @ja
footer: Position and momentum probability densities for an initial Gaussian distribution. From top to bottom, the animations show the cases
footer: Ω = ω/2
description: la principo, ke kvantuma stato ne povas havi kaj certan pozicion kaj certan movokvanton @eo
footer: . Note the tradeoff between the widths of the distributions.
image: Propagation of a de broglie plane wave.svg
label: Relación de indeterminación de Heisenberg @es
label: 불확정성 원리 @ko
subject: Category:Scientific laws
description: principio fondamentale della meccanica quantistica @it
description: квантова механіка @uk
subject: Category:Principles
description: مبدأ التشكيك و عدم اليقين البَّاري الخاص بالتتربب الإلكتروني ( الإلكترونات ) @ar
label: Heisenbergen ziurgabetasunaren printzipioa @eu
label: Principe d'incertitude @fr
width: 250
description: podstawowe prawo mechaniki kwantowej @pl
title: Proof of the Schrödinger uncertainty relation
description: Principio de la mecánica cuántica @es
label: Necerteca principo de Heisenberg @eo
description: fundamenteel principe in de kwantummechanica @nl
description: Fundamental principle in quantum physics
label: Princípio da incerteza de Heisenberg @pt
label: Heisenbergsche Unschärferelation @de
label: Prionsabal éiginnteachta Heisenberg @ga
image: Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_wide.gif
proof: The derivation shown here incorporates and builds off of those shown in Robertson, Schrödinger and standard textbooks such as Griffiths. For any Hermitian operator , based upon the definition of variance, we have we let and thus Similarly, for any other Hermitian operator in the same state for The product of the two deviations can thus be expressed as In order to relate the two vectors and , we use the Cauchy–Schwarz inequality which is defined as and thus Equation can be written as Since is in general a complex number, we use the fact that the modulus squared of any complex number is defined as , where is the complex conjugate of . The modulus squared can also be expressed as we let and and substitute these into the equation above to get The inner product is written out explicitly as and using the fact that and are Hermitian operators, we find Similarly it can be shown that Thus, we have and We now substitute the above two equations above back into Eq. and get Substituting the above into Equation we get the Schrödinger uncertainty relation This proof has an issue related to the domains of the operators involved. For the proof to make sense, the vector has to be in the domain of the unbounded operator , which is not always the case. In fact, the Robertson uncertainty relation is false if is an angle variable and 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. This issue can be overcome by using a variational method for the proof, or by working with an exponentiated version of the canonical commutation relations. Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operators and are self-adjoint operators. It suffices to assume that they are merely symmetric operators.
id: p/u095100
description: principi de la física quàntica @ca
footer: Ω = ω
subject: Category:Mathematical physics
label: Zasada nieoznaczoności @pl
description: 量子力學概念 @zh
description: Inégalités mathématiques de la mécanique quantique @fr
label: Принцип невизначеності @uk
align: right
label: 不确定性原理 @zh
description: Предел точности измерения величин квантовой системы, открытый в 1927 году @ru
label: Принцип неопределённости @ru
label: Osäkerhetsprincipen @sv