What is Uncertainty relation: Definition and 44 Discussions

In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions.
Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.
Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

where ħ is the reduced Planck constant, h/(2π).
Historically, the uncertainty principle has been confused with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system, that is, without changing something in a system. Heisenberg utilized such an observer effect at the quantum level (see below) 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. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology. 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. 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 superconducting or quantum optics systems. Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers.

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  1. D

    I Time-energy uncertainty relation

    Hi I have just been looking at the derivation of the uncertainty relationship for non-commutating operators. I have come across the following quote in Quantum Mechanics by Mandl regarding the time-energy relationship. "Time is not an operator ; it is an ordinary parameter which commutes with...
  2. Yan Campo

    I Robertson uncertainty relation for the angular momentum components

    I'm studying orbital angular momentum in the quantum domain, and I've come up with the Robertson uncertainty relation for the components of orbital angular momentum. Therefore, I read that it is necessary to pay attention to the triviality problem, because in the case where the commutator is...
  3. Wannabe Physicist

    Exploring the Time-Energy Uncertainty Relation to Solve a Schrodinger Equation

    I am guessing time-energy uncertainty relation is the way to solve this. I solved the Schrodinger equation for both the regions and used to continuity at ##x=-a, 0,a## and got ##\psi(-a<x<0) = A\sin(\kappa(x+a))## and ##\psi(0<x<a) = -A\sin(\kappa(x-a))## where ##\kappa^2 = 2mE/\hbar^2##...
  4. Jamister

    A Heisenberg Uncertainty Relation for mixed states

    How do you prove Heisenberg uncertainty relations for mixed states (density matrix), only from knowing the relation is true for pure states?
  5. S

    I Uncertainty relation derivation

    Hey guys , my lecturer introduced a new concept with reference to the commutation of two operators.He claimed that if two commutators commute then they can be simultaneously measured.I can clearly see how this works.He then went on and state if they don't commute they can't simultaneously be...
  6. Derek P

    I The inequality in the Heisenberg uncertainty relation

    I was musing about why the HUP is an inequality. If you analyse a wave packet the spatial frequency spectral width is inversely proportional to the spatial width. So there should be an equality such as Heisenberg's equation 3 in this paper. Has anyone got a simple explanation of where the...
  7. A. Neumaier

    I On the Heisenberg uncertainty relation

    Are there fundamental limits on the accuracy for measuring both position ##q## at time ##t## and momentum ##p## at time ##t+\Delta t##, with tiny ##\Delta t##? If yes, why? If no, why can't one then measure (in principle) both ##q## and ##p## arbitrarily well at the same time ##p## (which is...
  8. A. Neumaier

    B Position of a harmonic oscillator

    Here is the new thread. Please justify your criticism of my statement.
  9. entropy1

    B Thought experiment violates Heisenberg? (of course not)

    Suppose we have an elementary double-slit experiment: A laser fires individual photons through a double slit at a detectionscreen made of atoms. As we fire photons, an interference pattern emerges, exposing the momentum of the photons (the frequency of the laser). So, we have registration of...
  10. phys-student

    Quantum harmonic oscillator, uncertainty relation

    Homework Statement Consider a particle with mass m oscillates in a simple harmonic potential with frequency ω. The position, x, and momentum operator, p, of the particle can be expressed in terms of the annihilation and creation operator (a and a† respectively): x = (ħ/2mω)^0.5 * (a† + a) p =...
  11. entropy1

    Difference/similarity between superposition and uncertainty

    What is the relation between "superposition" and "the Heisenberg uncertainty relation"?
  12. weezy

    How is the uncertainty relation preserved in this experiment

    For an electron can I not do the following to determine both the position and momentum? I take a screen with a small hole and I eventually make the hole smaller and smaller. Cathode rays emitted will hence get diffracted after passing through the hole making momentum more and more uncertain...
  13. A

    How can the uncertainty relation be written as such

    Homework Statement Show that the uncertainty relation can be written as Δλ Δx >= λ^2 /4π Homework EquationsThe Attempt at a Solution Ok the uncertainty relation is ΔpΔx >= h/2π , also p = h/λ , so substituting that I have Δh/λ Δx >= h/2π , then divide both sides by h, and multiply both sides...
  14. A

    What is the Minimum Energy of a One-Dimensional Linear Oscillator?

    Homework Statement The (classical) energy of one-dimensional linear oscillator is a) show, using the uncertainty relation, that the energy can be written as b) Show that the minimum energy of the oscillator is Where Homework Equations Δp Δx >= ħ/2 p ≈ ħ/2x The Attempt at a Solution I'm...
  15. snoopies622

    Can one derive the photon number / phase uncertainty relation

    In an earlier thread of mine, another physics forums member was nice enough to point out that there is an uncertainty relation between photon number and wave phase for light. https://www.physicsforums.com/threads/is-there-a-frequency-eigenstate-for-light.727141/ Now I am wondering, where does...
  16. T

    A time-energy uncertainty relation

    I am reading this: http://arxiv.org/pdf/quant-ph/0609163.pdf And Demystifier claims that "The time-energy uncertainty relation is not fundamental" However the proof is done in non-relativistic QM, where t and x are treated differently. My question is, what's about relativistic QM?
  17. A

    General Uncertainty Relation -- Why drop anticommutator?

    In the derivation of the generalized uncertainty principle (as pgs 1-2 of here), there is an anticommutator term that is dropped at the end, leaving just the commutator part...this is said to "strengthen" the relation, as both terms are positive. I don't understand this. So we basically have...
  18. Greg Bernhardt

    What is uncertainty relation

    Definition/Summary One of the most asked questions is concerning how to derive the Heisenberg Uncertainty Relation. Starting from almost basic concepts of Quantum Mechanics, a derivation is given here. Some details are left as minor exercises for the interested reader. The derivation...
  19. MrRobotoToo

    Meaning of the Energy-Time Uncertainty Relation

    I'd like to know what exactly it's telling us. Does it mean that the more accurately we measure the energy of a system the less accurately we know for how long the system has been in that range of energies? Or does it mean that the more accurately energy is measured the less accurately we know...
  20. K

    The Uncertainty Relation for Position and Momentum

    Homework Statement The position of a 60-gram golf ball sitting on a tee is determined within +- 1μm. What is its minimum possi- ble energy? Moving at the speed corresponding to this kinetic energy, how far would the ball move in a year?Homework Equations ## K\geq\dfrac {\hbar ^{2}} {2ma^{2}}...
  21. vanhees71

    Heisenberg-Robertson uncertainty relation vs. noise-disturbance measures

    Finally I found the time to write my account on the interpretation of the Heisenberg uncertainty principle vs. the question whether it can be interpreted as Heisenberg did in his very first paper on the subject. Although it is well known that this interpretation is not compatible with quantum...
  22. H

    Energy-Time Uncertainty Relation

    I was trying to Go from the uncertainty principle to its energy-time counter part. i know the maths is a bit off,but the idea is correct? dx=position p=momentum e=energy \upsilon=frequency \lambda=wavelength c=velocity of electromagnetic radiations dt=time now , \lambda=h/p....(i)...
  23. S

    Rejecting the Semiclassical Bohr Model: Examining the Uncertainty Relation

    "Show that the uncertainty relation forces us to reject the semiclassical Bohr [...]" Homework Statement The problem along with the solution is attached as TheProblemAndSolution.jpg. Homework Equations Uncertainty principle/relation. The Attempt at a Solution Why is it the consideration of...
  24. F

    Uncertainty Relation between Lx and Ly

    There is an uncertainty relation between the x component and the y component of the angular momentum L of a particle, because [Lx, Ly] = i\hbarLz which is not 0. But what happens when Lz does equal 0? Would we in principle be able to measure both the x and y components of the angular...
  25. A

    Timelike Curves leads to violation of heisenberg uncertainty Relation

    General Relatitivity predicts Timelike curves and there are nonlinear extensions of mechanics which resolve the paradoxical aspects of CTC's *i.e. Time Travel, on the other hand Hawking proposed a conjeture to rule out CTCs, the Chronology Protection Conjecture* there are a class of Timelike...
  26. G

    Does the uncertainty relation apply to the past?

    Can anybody clear this up for me? In his Chicago lectures in 1930, Heisenberg is quoted as saying “The uncertainty relation does not hold for the past…If the velocity of the electron is at first known, and the position then exactly measured, the position of the electron for times previous...
  27. M

    Uncertainty relation between number of photons in an EM field and its phase

    Derive the relation Δn.Δσ ≥ 1/2 where n is number of photons in an EM field and σ is phase Using heisenburgs uncetertainty principle? Tried subbing in frequency into heisenburgs uncertainty principle to get to the number of photons and to get rid of mometum is this the right line? Dont...
  28. N

    Variable pairs in uncertainty relation?

    I have been trying to figure out other pairs of variables in Heisenberg's uncertainty relationship apart from the well known position-momentum and time-energy pairs. I am particularly interested in electric fireld strength and magnetic field strenght. The reason for my interest is that if I...
  29. C

    Time-Energy Uncertainty relation

    Homework Statement So I have this homework question and the answer that was given to me to check the answer that I have does not match with my answer. The question I have to complete is shown below: The answer given to me is D, 6.6eV Homework Equations \Deltat\DeltaE = h/2...
  30. A

    Uncertainty relation demonstration

    HI, I will not used the template provided since this is not a textbook problem. It's a problem I have with a demonstration in Quantum Mechanics Vol 1, Cohen-Tannoudji " Complement C III [Q,P] = i \hbar Consider the ket: |\phi \rangle = (Q +i\lambda P)|\psi \rangle where \lambda is an...
  31. F

    Uncertainty relation: minimum value

    Homework Statement Show that the smallest possible uncertainty in the position of an electron whose speed is given by \beta = v/c is \Delta x_{min} = \frac{h}{4 \pi m_0 c}\sqrt{1-\beta^2} The Attempt at a Solution Since \Delta x \Delta p \geq \frac{\hbar}{2} , we see that \Delta x_{min}...
  32. P

    The uncertainty relation and density of states

    Homework Statement I am trying to understand how we go about calculating the density of states in situations where the available quantum states are continuous, e.g. electrons in a white dwarf. I am happy to accept the uncertainty relation (we learned to derive it as the product of the...
  33. L

    Proof in heisenbergs uncertainty relation involving bra-ket

    hey, can someone show me the step between these two lines of equations please: (\Delta A)^2=<\psi|A^2|\psi>-<\psi|A|\psi>^2 =<\psi|(A-<A>)^2|\psi> where A is an operator and \psi is the wavefunction and <A> is the expectation value of A
  34. R

    Experiments for time-energy uncertainty relation

    Are there experiments which you can claim that they are the evidence of the time-energy uncertainty relation? Or can you write which experiments can be directly related to this uncertainty relation? i.e A beta decay or a strange effect in (quantum) optics... everything you have in mind...
  35. K

    Confused again by uncertainty relation

    Say we have two non-commutative operator A, B. Now I have prepared identical systems in eigenstate of A, then I measure the observable of A, and then B immediately after. Then I must have delta A=0, so no matter what delta B is, the product is 0, seems to violate the uncertainty relation...
  36. O

    Proving the Heisenberg Uncertainty Relation | Homework Equations Included

    Homework Statement Could someone please have a look at this? I am to show that from the inequation \langle\left \psi | \hbar^2D^2 | \psi\right\rangle + mk\langle \left\psi | x^2 | \psi\right\rangle\geq\hbar\sqrt{mk} you can get the Heisenberg uncertainty relation \langle\psi |...
  37. H

    Operators, normalised eigenstates and the generalised uncertainty relation

    Homework Statement Hi guys! Many time reader, first time poster... I've struggled big time with the following. Any advice at all would be great. I'm so muddled, it's just not funny any more... (plus I'm not really familiar with who to write the mathematic script so please be patient) I...
  38. H

    Lifetime & uncertainty relation w.r.t. particles

    Can anyone explain how to interpret Heisenberg's relation Delta(E)*Delta(t)>=hbar in case of "annihilation" and or "creation" of (elementary) particles: 1) in case of virtual particles 2) in case of antiparticles ? Is it eventually useful, if I may say so, to discriminate between the...
  39. Reshma

    General Uncertainty Relation between 2 operators

    The general uncertainty relation between two observables A and B. (\Delta A)^2(\Detla B)^2 \geq -{1\over 4}<[A, B]>^2 I have to prove the above relation using the definition of expection values etc. The reference I use (Liboff) have this relation given as an exercise. But Gasiorowicz's book...
  40. N

    Heisenberg's uncertainty relation not invariant ?

    I'm not sure whether this has been discussed before. If one just looks at Heisenberg's uncertainty relation (energy-time), one easily sees that this is not Lorentz invariant. Even very simple results, such as the energy of a particle in a potential well seems not to transform according to the...
  41. quasar987

    Time-energy uncertainty relation

    How should \Delta E \Delta t \geq \hbar/2 be interpreted? When does it apply? Delta t refers to the time of what? etc. Thx.
  42. M

    Uncertainty relation and absolute zero.

    Is there a certain fixed degree of uncertainty? Like for example, we can't reach absolute zero because we would know both the momentum and position of the particle. But if it is say, 3-4 K above absolute zero, there would be a really small degree of uncertainty involved, because its quite...
  43. R

    Energy-time uncertainty relation

    Is there an operator which corresponds to time whose commutator with the Hamiltonian equals ih? I mean, when you collapse the wave function by taking an energy measurement, then at that instant the uncertainty in energy is 0. But then as times goes on the state will be that energy eigenstate and...
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