Physics
Uncertainty Relations in Quantum Mechanics
Uncertainty relations in quantum mechanics, such as the Heisenberg uncertainty principle, describe the limitations on the precision with which certain pairs of physical properties of a particle can be simultaneously known. These relations imply that the more precisely one property is measured, the less precisely the other can be known, highlighting the inherent uncertainty in quantum systems.
Written by Perlego with AI-assistance
Related key terms
1 of 5
12 Key excerpts on "Uncertainty Relations in Quantum Mechanics"
- Stefano Spezia(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
SECTION II: UNCERTAINTY PRINCIPLE AND MEASUREMENT IN QUANTUM MECHANICS REFORMULATING THE QUANTUM UNCERTAINTY RELATION 4 CHAPTER CITATION : Li, J.-L. and Qiao, C.-F., “Reformulating the Quantum Uncertainty Relation”, Scientific Reports (2015), https://doi.org/10.1038/srep12708 COPYRIGHT: © 2015 The Authors. Published by Macmillan Publishers Limited, part of Springer Nature. This is an open access article under the CC BY license (http://creativecom-mons.org/licenses/by/4.0). Jun-Li Li 1 and Cong-Feng Qiao 1,2 1 Department of Physics, University of the Chinese Academy of Sciences, YuQuan Road 19A, Beijing 100049, China. 2 CAS Center for Excellence in Particle Physics, Beijing 100049, China. ABSTRACT Uncertainty principle is one of the cornerstones of quantum theory. In the literature, there are two types of uncertainty relations, the operator form concerning the variances of physical observables and the entropy form related to entropic quantities. Both these forms are inequalities involving Basic Quantum Mechanics for Electrical Engineering 60 pairwise observables, and are found to be nontrivial to incorporate multiple observables. In this work we introduce a new form of uncertainty relation which may give out complete trade-off relations for variances of observables in pure and mixed quantum systems. Unlike the prevailing uncertainty relations, which are either quantum state dependent or not directly measurable, our bounds for variances of observables are quantum state independent and immune from the “triviality” problem of having zero expectation values. Furthermore, the new uncertainty relation may provide a geometric explanation for the reason why there are limitations on the simultaneous determination of different observables in N-dimensional Hilbert space. INTRODUCTION The uncertainty principle is one of the most remarkable characteristics of quantum theory, which the classical theory does not abide by.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 5 Uncertainty Principle In quantum mechanics, the Heisenberg uncertainty principle states by precise inequalities that certain pairs of physical properties, such as position and momentum, cannot be simultaneously known to arbitrarily high precision. That is, the more precisely one property is measured, the less precisely the other can be measured. The principle states that a minimum exists for the product of the uncertainties in these properties that is equal to or greater than one half of the reduced Planck constant (ħ = h/2π). Published by Werner Heisenberg in 1927, the principle means that it is impossible to determine simultaneously both the position and momentum of an electron or any other particle with any great degree of accuracy or certainty. Moreover, his principle is not a statement about the limitations of a researcher's ability to measure particular quantities of a system, but it is a statement about the nature of the system itself as described by the equations of quantum mechanics. In quantum physics, a particle is described by a wave packet, which gives rise to this phenomenon. Consider the measurement of the position of a particle. It could be anywhere. The particle's wave packet has non-zero amplitude, meaning the position is uncertain – it could be almost anywhere along the wave packet. To obtain an accurate reading of position, this wave packet must be 'compressed' as much as possible, meaning it must be made up of increasing numbers of sine waves added together. The momentum of the particle is proportional to the wavenumber of one of these waves, but it could be any of them. So a more precise position measurement–by adding together more waves– means the momentum measurement becomes less precise (and vice versa). The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength (and therefore an indefinite momentum).
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 5 Uncertainty Principle In quantum mechanics, the Heisenberg uncertainty principle states by precise inequalities that certain pairs of physical properties, such as position and momentum, cannot be simultaneously known to arbitrarily high precision. That is, the more precisely one property is measured, the less precisely the other can be measured. The principle states that a minimum exists for the product of the uncertainties in these properties that is equal to or greater than one half of the reduced Planck constant (ħ = h/2π). Published by Werner Heisenberg in 1927, the principle means that it is impossible to determine simultaneously both the position and momentum of an electron or any other particle with any great degree of accuracy or certainty. Moreover, his principle is not a statement about the limitations of a researcher's ability to measure particular quantities of a system, but it is a statement about the nature of the system itself as described by the equations of quantum mechanics. In quantum physics, a particle is described by a wave packet, which gives rise to this phenomenon. Consider the measurement of the position of a particle. It could be anywhere. The particle's wave packet has non-zero amplitude, meaning the position is uncertain – it could be almost anywhere along the wave packet. To obtain an accurate reading of position, this wave packet must be 'compressed' as much as possible, meaning it must be made up of increasing numbers of sine waves added together. The momentum of the particle is proportional to the wavenumber of one of these waves, but it could be any of them. So a more precise position measurement–by adding together more waves– means the momentum measurement becomes less precise (and vice versa). The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength (and therefore an indefinite momentum).
eBook - PDF- Charles G. Wohl(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
7. UNCERTAINTY RELATIONS. SIMULTANEOUS EIGENSTATES 7.1. Heisenberg Uncertainty Relations 7.2. The Schwarz Inequality 7.3. Proof of Uncertainty Relations 7.4. Fourier Transforms. Momentum Space 7.5. Time and Energy 7.6. When Operators Commute Problems Most of this chapter is about whether the values of two physical properties of a particle (observables) can be precisely known at the same instant. That depends on whether the corresponding operators commute or not. When two operators do not commute, there is an “uncertainty relation,” which sets a limit on how well the values can be simultaneously known. The famous example is Δx Δp x ≥ 1 2 ¯ h, where x and p x are the position and momentum of a particle along a rectangular axis, and Δx and Δp x are the root-mean-square (standard- deviation) measures of the uncertainties in the values. Those uncertainties depend on the state of the system. For example, as n increases, the oscillator energy eigenfunctions ψ n (x) spread wider in x, and both Δx and Δp x are larger. The absence of exact knowledge of the state of a system is not somehow due to short- comings of our experimental arrangements. Werner Heisenberg put it perfectly: “... in the strong formulation of the causal law ‘If we know exactly the present, we can predict the future’ it is not the conclusion but rather the premise which is false. We cannot know, as a matter of principle, the present in all its details.” In Section 3, we derive the uncertainty relation for the observables A and B from the commutator [ ˆ A, ˆ B] of their operators. Then Δx Δp x ≥ 1 2 ¯ h follows at once from [ˆ x, ˆ p x ] = i¯ h. In the next chapter, we come to the more complicated [ ˆ L x , ˆ L y ] = i ˆ L z (and cyclic permutations of x, y, and z), where ˆ L x , ˆ L y , and ˆ L z represent the components of the orbital angular momentum L of a particle or system.
eBook - PDF- Don B Lichtenberg(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
What Heisenberg’s uncertainty relations tell us is that no matter how good our measuring aparatus is, it is impossible to measure, for example, both the position and the momentum of a particle to arbitrary accuracy. The product of the uncertainty in the measure-ments of the position and the momentum must be at least as great as Planck’s constant divided by 4 π . The better the measurement of 13.3. HEISENBERG’S UNCERTAINTY RELATIONS 155 the position, the worse will be the measurement of the momentum, and vice versa. Likewise, the more precisely we measure the time at which we observe a particle, the poorer will be our measurement of the energy of the particle and vice versa. In Newtonian mechanics it was assumed that we could measure both the position and the velocity of a particle with arbitrary accu-racy. We see that the theory of quantum mechanics says it isn’t so. Why, then, did the Newtonian assumption go unchallenged for more than 200 years? The answer is that Planck’s constant is very small and the mass of a macroscopic particle is very large (compared to the mass of an electron). The mass of the particle is relevant because momentum is equal to mass times velocity ( p = mv ). Suppose the position of a particle is measured with only a small error. Then, by Heisenberg’s uncertainty principle, the error in the momentum must be large. But if the mass of the particle is large, then, because of the equation p = mv , even a small error in the velocity leads to a large error in the momentum because the small error in the velocity is mul-tiplied by a large quantity: the mass. For a large enough mass, the minimum error in the velocity will usually be too small to observe, so that Newtonian mechanics becomes an excellent approximation to quantum mechanics. The situation is different for a particle of small mass, like the elec-tron. The mass of the electron m e has been measured to be m e = 9 .
eBook - PDF- Ivan Stanimirovi?, Olivera M. Stanimirovi?, Ivan Stanimirović, Olivera M. Stanimirović(Authors)
- 2020(Publication Date)
- Arcler Press(Publisher)
This result, like so many others in quantum mechanics, only affects the subatomic physical chemistry, because Planck’s constant is quite small, in a macroscopic world quantum uncertainty is negligible, and remain valid relativistic theories, such as Einstein. Introduction 23 In quantum mechanics, particles do not follow defined paths, you cannot know the exact value of the physical variables that explain the state of motion of a particle, only one statistic distribution, so neither can know the trajectory of a particle. But instead, can itself be said that there is a certain probability that a particle is in a particular region of space at a given time. It is often said that scientific determinism vanishes with the probabilistic nature of quantum, but there are different ways of interpreting quantum mechanics, and for example, Stephen Hawking says that quantum mechanics itself is deterministic, it being possible for his alleged indeterminacy is because actually there are no positions or velocities of particles, but all are waves. Thus, quantum physicists and chemists try to insert the waves within our previous ideas of positions and velocities. The “uncertainty principle” greatly influenced the physical and philosophical thought of the time. Often read that the principle of uncertainty clears all the certainties of nature, implying that science does not know or will ever know where you are going, because the dependent scientific knowledge of the unpredictability of the universe, where the cause-effect does not always go hand in hand. ( ) 2 2 2 2 2 2 2 2 0 E V X y x h δ ψ δ ψ δ ψ δ ψ ψ ε ε ε + + + − = Bohr had postulated a model working perfectly for hydrogen atom, which necessitated an urgent correction model. It was Somerfield who modified the Bohr model to deduce that each energy level were sublevels, adding explained elliptical orbits and using relativity.
eBook - PDFThe Quantum Relations Principle
Managing our Future in the Age of Intelligent Machines
- Hardy F. Schloer, Mihai I. Spariosu(Authors)
- 2016(Publication Date)
- V&R Unipress(Publisher)
The uncertainty principle of quantum physics is worth further explanation, because it reflects a more general principle of certain algebras (in QR terms, arithmetics). As mentioned earlier, QM uses the concept of a probability density. It can be thought of as being denser where things are likely, and less dense where things are unlikely. Generally, the probability density is given by some function, called the probability density function, or simply the pdf. Scientists who study QM or the mathematics of probability have devised all sorts of ways of calculating the pdf for starting in one particular state of a particular system, then after some time, reaching another particular state. None of these ways is important here, but once the pdf is known, you can calculate lots of other interesting things that describe states of the system and their evolution over time. We shall now break our promise of avoiding mathematical formulae. But, the non-mathematical reader can simply skip this section and move to the next, keeping in mind only that Quantum Relations, like QM, is a probabilistic theory with some basic, irreducible quanta. The consequence of this, from a purely mathematical viewpoint, is that both systems have built-in and unavoidable uncertainty. For QM, the uncertainties are tiny compared to real-world events, and they are unlikely to be observed by anyone other than high-energy phys- icists. QM uncertainties do not explain psychic phenomena, they do not explain magic, and they do not explain why people forget their house keys. Furthermore, QR uncertainties are not caused by QM uncertainty and, except for some math- ematical parallelism, they are probably not related to it. But in QR, the calcu- Why is Quantum Relations Quantum? 38
eBook - PDF- Mohsen Razavy(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
(4.46) The converse of this result is also true, that is if two operators commute then they possess the same eigenvector. Let | n i be an eigenvector of A so that Uncertainty Principle 91 A | n i = a n | n i , then AB | n i = A ( B | n i ) and B ( A | n i ) = B ( a n | n i ). Therefore if AB = BA we have A ( B | n i ) = a n ( B | n i ) , (4.47) or B | n i is an eigenvector of A . This can also be true if the ket B | n i is propor-tional to | n i or B | n i = b n | n i . (4.48) Here we have implicitly assumed that there is no degeneracy, but if there are degenerate states the theorem remains valid. 4.1 The Uncertainty Principle In his seminal paper of 1927, Heisenberg rejected the idea of strict observabil-ity of the trajectory of an electron (position as well as momentum) as it was assumed in Bohr’s model. Instead he considered the observable trajectory (e.g. in the Wilson cloud chamber) as a discrete sequence of imprecisely defined po-sitions. Soon after the publication of his paper, most of the physicists regarded the uncertainty relations as an integral part of the foundation of quantum the-ory [7]. In classical mechanics, according to Laplace, if one knows the exact po-sition and velocity of the particle at a given time, then one can predict the position and the velocity of the particle at any future time provided that all of the forces acting on the particle is known precisely [8]. This is one of the simplest and widely-accepted principles of causality in mechanics. Now the un-certainty principle invalidates this principle and replaces it with the following observation: In quantum theory we can calculate only a range of possibilities for the position and velocity of the electron for later times, one of which will result from the motion of a given electron. However, as was observed by Heisenberg, the predictions of quantum mechanics are statistical in nature [9].
eBook - PDFQuantum Mechanics
Foundations and Applications
- Donald Gary Swanson(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
In quantum mechanics, the uncertainty principle limits the exactness of our knowledge. It should be understood that although the equations of quan-tum mechanics are deterministic in nature (so that for a particular initial condition, the system at a later time is completely determined), the initial conditions cannot be determined exactly, so the outcome can only be stated as a probability. In view of this, the postulates describe the state of a system in terms of probabilities and probability amplitudes. The first postulate puts this notion into quantitative form. 1.4.1 Postulate 1 There exist two complex probability amplitudes (also called wave func-tions ), Ψ( q j , t ) and Φ( p j , t ) that completely define the state of a quantum-mechanical system in the following way: If at time t , the coordinates of the The Foundations of Quantum Physics 25 system are measured, the probability that these will be found to lie within the ranges q j to q j + d q j is W q ( q j , t ) d q 1 · · · d q N = Ψ * ( q j , t )Ψ( q j , t ) d q 1 · · · d q N , (1.22) while if, instead, the momenta were measured at time t , the probability that they would be found to lie within the interval between p j and p j + d p j is W p ( p j , t ) d p 1 · · · d p N = Φ * ( p j , t )Φ( p j , t ) d p 1 · · · d p N . Since each coordinate must have some probability of being found somewhere , it is also usually required that the total integrated probability be unity, or that Ψ * Ψ d N q = 1 , and Φ * Φ d N p = 1 , where d N q = d q 1 · · · d q N and a single sign represents an integration over all of the variables over their entire range. Occasionally, the integral will be set to some arbitrary value, or only relative probabilities may be given where the ratio of two unbounded integrals is finite.
eBook - ePubV.A. Fock - Selected Works
Quantum Mechanics and Quantum Field Theory
- L.D. Faddeev, L.A. Khalfin, I.V. Komarov, L.D. Faddeev, L.A. Khalfin, I.V. Komarov(Authors)
- 2004(Publication Date)
- CRC Press(Publisher)
OCK AND N. KRYLOV (Received 29 May 1946)JETP 17, N 2, 93, 1947J. Phys. USSR 11, N 2, 112, 1947 (English version)Introduction
The physical meaning of the uncertainty relation between time and energy appears to be not completely clear until now. This fact is not surprising since the interpretation of the time–energy relation is much more difficult than the interpretation of the similar relation between the coordinate and momentum. Indeed, the latter relation is easily derived from the quantum-mechanical formalism and does not require the examination of the course of the physical process in time. The relation between time and energy, however, essentially requires such an examination, and there arises a question whether Schrödinger’s equation may be used or not in deriving this relation, the latter possibility being not excluded since the relation may be interpreted to correspond to a measurement act, which does not obey Schrödinger’s equation.L. Mandelstam and Ig. Tamm in their recent paper (The Uncertainty Relation between Energy and Time in Non-Relativistic Quantum Mechanics) [1] made a very interesting attempt to derive the abovementioned uncertainty relation from the Schrödinger equation.The aim of the present note is to analyze the relation derived by these authors. At first we shall consider the uncertainty relation in its usual (Bohr’s) interpretation and compare it with the relation derived by Mandelstam and Tamm. We shall find that the meaning and regions of application of these two relations are entirely different. In the last section we shall establish a connection between the law of decay of an almost stationary state and the energy distribution function in this state and consider the question on the practical applicability of the Mandelstam– Tamm relation to this problem.
eBook - PDF- David J. Griffiths, Darrell F. Schroeter(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
But these photons impart to the par- ticle a momentum you cannot control. You now know the position, but you no longer know the momentum. Bohr’s famous debates with Einstein include many delightful examples, showing in detail how experimen- tal constraints enforce the uncertainty principle. For an inspired account see Bohr’s article in Albert Einstein: Philosopher-Scientist, edited by Paul A. Schilpp, Open Court Publishing Co., Peru, IL (1970). In recent years the Bohr/Heisenberg explanation has been called into question; for a nice discussion see G. Brumfiel, Nature News https://doi.org/10.1038/nature.2012.11394. 108 CHAPTER 3 Formalism make the second measurement without disturbing the state of the particle (the second col- lapse wouldn’t change anything, in that case). But this is only possible, in general, if the two observables are compatible. ∗ Problem 3.14 (a) Prove the following commutator identities: ˆ A + ˆ B , ˆ C = ˆ A, ˆ C + ˆ B , ˆ C , (3.64) ˆ A ˆ B , ˆ C = ˆ A ˆ B , ˆ C + ˆ A, ˆ C ˆ B . (3.65) (b) Show that x n , ˆ p = i nx n−1 . (c) Show more generally that f (x ), ˆ p = i d f dx , (3.66) for any function f (x ) that admits a Taylor series expansion. (d) Show that for the simple harmonic oscillator ˆ H , ˆ a ± = ±ω ˆ a ± . (3.67) Hint: Use Equation 2.54. ∗ Problem 3.15 Prove the famous “(your name) uncertainty principle,” relating the uncer- tainty in position ( A = x ) to the uncertainty in energy ( B = p 2 /2m + V ) : σ x σ H ≥ 2m | p| . For stationary states this doesn’t tell you much—why not? Problem 3.16 Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if ˆ P and ˆ Q have a complete set of common eigenfunctions, then ˆ P , ˆ Q f = 0 for any function in Hilbert space.
- Mohammad Reza Pahlavani(Author)
- 2015(Publication Date)
- IntechOpen(Publisher)
Lett. Vol. B 452: 39-44. [37] Scardigli, F. & Casadio, R. (2003). Generalized uncertainty principle, extra dimensions and holography, Classical and Quantum Gravity, Class. Quantum Grav. Vol. 20: 3915-3926. [38] Singh, J. (1997). Quantum Mechanics-Fundamentals and Applications to Technology (New York: A Wiley Interscience). Selected Topics in Applications of Quantum Mechanics 98 Chapter 4 Unification of Quantum Mechanics with the Relativity Theory, Based on Discrete Conservations of Energy and Gravity Aghaddin Mamedov Additional information is available at the end of the chapter http://dx.doi.org/10.5772/59169 1. Introduction The history of physics has two great revolutionary theories, such as relativity and quantum physics. However, the new discoveries of the particle physics do not fit with the principles of both theories. One of the main questions is how to explain the break of symmetry in proton-antiproton collision experiments and formation of more matter particles than that of anti-matter ingredients. The theory, which can explain this phenomenon, should answer to the question how matter in the beginning of universe formed and what is the space-time structure of the universe. In accordance with the modern physics, the small-scale experiments of particles physics have to be described by the Standard Model of quantum mechanics. However, the phenomenon of matter/antimatter symmetry breaking appearing at subatomic scale requires formulation of the new dynamical laws. Presently it is not clear that the mystery of the small-scale dynamics is due to the incompleteness of the mathematic formulation of dynamics of physical events at small scale or to the change of mathematics. The problem is that our present knowledge on mathematical description of change at small scale of space-time frame and physical theories do not distinguish what special features should have the initial state (position) of universe.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
