Chemistry
Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and momentum of a particle. In the context of chemistry, this principle has implications for the measurement and understanding of atomic and subatomic particles. It fundamentally challenges the idea of precise measurement at the quantum level.
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11 Key excerpts on "Heisenberg Uncertainty Principle"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 4 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. Published by Werner Heisenberg in 1927, the principle implies that it is impossible to determine simultaneously both the position and the momentum of an electron or any other particle with any great degree of accuracy or certainty. This is not a statement about researchers' ability to measure the quantities. Rather, it is a statement about the system itself. That is, a system cannot be defined to have simultaneously singular values of these pairs of quantities. 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). Historical introduction Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics. 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 motion was not precise at the quantum level, and electrons in an atom did not travel on sharply defined orbits. Rather, the motion was smeared out in a strange way: the Fourier transform of time only involving 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.
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- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
This correspondence principle is now generally accepted. It suggests the rules of classical mechanics are an approximation of the rules of quantum mechanics for systems with very large energies. Quantum mechanics describes both the microscopic and macroscopic world, but classical mechanics describes only the latter. 312 Chapter 7 | Quantum Mechanics This OpenStax book is available for free at http://cnx.org/content/col12067/1.4 7.2 | The Heisenberg Uncertainty Principle Learning Objectives By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg’s uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x-direction. The particle moves with a constant velocity u and momentum p = mu . According to de Broglie’s relations, p = ℏk and E = ℏω . As discussed in the previous section, the wave function for this particle is given by (7.14) ψ k (x, t) = A[cos(ωt − kx) − i sin(ωt − kx)] = Ae −i(ωt − kx) = Ae −iωt e ikx and the probability density | ψ k (x, t) | 2 = A 2 is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has definite values of wavelength and wave number, and therefore momentum.
eBook - ePub- Chad Orzel(Author)
- 2010(Publication Date)
- Oneworld Publications(Publisher)
She trots over to the TV, and sticks her nose under the cabinet. “Here’s my bone!” She paws at it for a minute, and eventually succeeds in knocking it out from under the cabinet. “I have a bone!” she announces proudly, and begins chewing it noisily, the uncertainty principle forgotten.The Heisenberg Uncertainty Principle is probably the second most famous result from modern physics, after Einstein’s E = mc2 (the most famous result from relativity). Most people wouldn’t know a wavefunction if they tripped over one, but almost everyone has heard of the uncertainty principle: it is impossible to know both the position and the momentum of an object perfectly at the same time. If you make a better measurement of the position, you necessarily lose information about its momentum, and vice versa.In this chapter, we’ll describe how the uncertainty principle arises from the particle-wave duality we’ve already discussed. The uncertainty principle is often presented as a statement that a measurement of a system changes the state of that system, and in this form, references to quantum uncertainty turn up in all sorts of places, from politics to pop culture to sports.* Ultimately, though, uncertainty has very little to do with the details of the measurement process. Quantum uncertainty is a fundamental limit on what can be known, arising from the fact that quantum objects have both particle and wave properties.Uncertainty is also the first place where quantum physics collides with philosophy. The idea of fundamental limits to measurement runs directly counter to the goals and foundations of classical physics. Dealing with quantum uncertainty requires a complete rethinking of the basis of physics, and leads directly to the issues of measurement and interpretation in chapters 3 and 4
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 - ePubQuantum Mechanics I
The Fundamentals
- S. Rajasekar, R. Velusamy(Authors)
- 2022(Publication Date)
- CRC Press(Publisher)
5.3 ×10m. According to quantum mechanics this value is the most probable radius. This means in experiment, most trials will give a different value, larger or smaller than− 115.3 ×m.10− 11- An important implication of the uncertainty principle is that quantum systems do not follow classical trajectories which conflict with our everyday experience.
Another consequence is that quantum mechanical systems in their lowest energy state (ground state) could not be at rest. For example, the electron of an atom in its ground state could not be at rest. What will happen if it is at rest? In this case its velocity is exactly known to be zero and its position would be known precisely. This would violate the Heisenberg uncertainty relation.Supposeψ ( x )is a delta-function atx =. The position of the particle is determined exactly. What about the corresponding momentum wave function? The momentum wave functionx 0C ( k )is a sinusoidal function extending over all space. Thus, the momentum uncertainty is infinity. Similarly, ifψ ( x )is sinusoidal then the position uncertainty is infinity whereas because the corresponding momentum wave function is delta-function the uncertainty in momentum is zero.It gives a resolution limit for the determination of the spectrum of a signal in terms of the measurement [7 ]. It also states resolution limits for imaging systems in terms of their aperture size. It serves as the basis of one of the standard measure of laser beam quality [8 ]. The uncertainty principle indicates the existence of quantum fluctuations in physical states that cannot be controlled completely. Particularly, the fluctuations of energy and momentum go arbitrarily large when the space-time interval becomes shorter and shorter.The interaction between two particles occurs by means of exchange of a quantum of the interacting field. For example, Yukawa's strong nuclear force between nucleons is due to the exchange of pions. The mass of the exchanged quantum is found to have an inverse relation with the range of the interaction. This can be accounted only by the uncertainty relation. If R is the range and m
- Volodymyr Krasnoholovets(Author)
- 2017(Publication Date)
- Apple Academic Press(Publisher)
p ) . An experimenter cannot know both the position and the momentum at the same moment in time. Heisenberg noted that a selection from an abundance of possibilities takes place in quantum systems, which also puts a limitation on future outcomes. Heisenberg worked with a broadened wave packet with calculable probability-a typical approach to the description of quantum systems.Later on Kennard [128 ] derived a different formulation of the uncertainty principle, which was later generalized by Robertson [129 ]: σ (q )σ (p ) ≥ h /(4π ), i.e., one cannot suppress quantum fuctuations of both position σ (q ) and momentum σ (q ) lower than a certain limit simultaneously. The fuctuations exist within themselves without respect to whether the position q and the momentum p of the quantum system are measured or not. This inequality cannot foresee any behavior of the parameters q and p at the time of measurement. Nowadays it seems that Kennard’s formulation of the uncertainty principle is more oft en used.Other authors deriving the uncertainty relation(3.5)Δ x Δ p ≥1 2ħbased their consideration on the behavior of a wave packet of finite length (see, e.g., Fermi [130 ] who referred to the proof conducted by E. Persico — see Engl. translation: Persico, E. (1950), Fundamentals of Quantum Me chanics , Prentice-Hall, New York) or the representation of the wave ψ -function as a superposition of plane waves corresponding to the discrete spectrum (Born [90 ], p. 383), and so on.In such a manner the uncertainty principle is considered as a corollary of the wave-particle duality of nature when a canonical particle is called a wave-particle and then all the characteristics of classical waves are automatically attributed to the particle. However, a wave packet is not stable and dissipates over time. Hence the description of a particle by using a wave packet is an approximate description. De Broglie [111 ] in his book devoted to the uncertainty principle analyzed this topic as well as the principle of spectral expansion in detail. He showed that the uncertainty relation (3.5 ) is due to: (i) the attribution of a wave function ψ
eBook - PDF- Ian Duck, E C George Sudarshan(Authors)
- 2000(Publication Date)
- World Scientific(Publisher)
[Read 'The concept of a perfectly precise observation with no other affect is essential to the classical deterministic space-time description.'] Due to the limitation of Eqn2, however, this description is possible in quantum theory only if the restrictions of the Heisenberg Uncertainty Principle are taken into account. [Read '• • • this descrip-tion is impossible in quantum theory because the restrictions of the Heisenberg Uncertainty Principle must be taken into account.'] (20) It is instructive to compare the fundamental uncertainties of the quan-tum description of microscopic phenomena with the ordinary imprecision - due to imperfect measurements - inherent in any classical observation. Macroscopic phe-nomena are defined by repeated observations. [Bohr's recognition of this fact is basic to modern interpretations of quantum mechanics (see Chl5).] In classical theory, each succeeding observation improves our knowledge of the initial state of the system. But in quantum theory, the impossibility of neglecting the effect of the observation on the object means that every observation introduces a new uncontrollable uncer-tainty. It follows from the Heisenberg Uncertainty Principle that any measurement of the position of the particle inevitably results in a corresponding finite change in its momentum, and a sacrifice in the description of its dynamical behavior. Con-versely, the determination of its momentum always implies a loss in the knowledge of its position. This limitation on the description of atomic phenomena is an in-evitable consequence of the quantum postulate, and contrasts with the classical distinction between object and instrument which is inherent in our very idea of 400 100 Years of Planck's Quantum observation. [Read '• • • contrasts with our classical concept of a pre-existing [c-]reality for the object, independent of the apparatus, which is inherent in our pre-Heisenberg classical concept of observation.'] § 4.
eBook - ePubPhilosophy of Science
The Link Between Science and Philosophy
- Philipp Frank(Author)
- 2013(Publication Date)
- Dover Publications(Publisher)
9 The New Language of the Atomic World 1. Heisenberg’s Uncertainty Relation 1 The laws of motion for small particles are formulated 2 in such a way that they connect observable initial conditions with observable results; the laws say nothing about “moving” particles. The scientist has always felt the need for retaining the traditional laws of motion as long as possible. They have been absorbed by our common-sense language, and it is certainly very helpful to use this same language as much as possible. In using common-sense language, the imagination of the scientist works with more ease and detachment than in using an abstract language where every result must be found by step-by-step formal reasoning. If we again consider the swarm of particles passing through the diaphragm through two slits S 1 and S 2, we can describe the situation in the following way: A wave passes through both slits and produces interference on the other side. To say that one particle passes through both slits would be an odd way to speak about a small particle ; therefore, one would do better to say that the particle passes through the diaphragm, but that the exact location of its passing is described by an uncertainty a because the distance between the slits through which it passes is a. In Newtonian mechanics, the initial conditions of motion are the position (coordinate) and velocity (or momentum). If these data are given, we can predict the future motion from Newton’s laws of motion
eBook - PDF- Belal E. Baaquie, Frederick H. Willeboordse(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
Hence we have x : (24.14) In the process of measurement, we have to scatter off the particle at least one 528 Exploring Integrated Science photon with wavelength . The photon, from Equation 23.8, is carrying momentum given by p D h= , which it will impart to the particle; hence, the change in the momentum of the particle due to the photon bouncing off it is given by p D h : (24.15) From Equations 24.14 and 24.15, we obtain xp h: In other words, the accuracy of the measurement that we make of the particle’s position and momentum is intrinsically limited; the least disturbance to the particle that we can cause by a measurement is to probe the particle with one photon , and it leads to the uncertainty derived above. A more precise statement is the following: xp „ 2 W Heisenberg’s Uncertainty Principle : (24.16) Note that the Heisenberg Uncertainty Principle states that if one knows the position of a particle exactly, that is, x D 0 , then p D 1 meaning that we have no information regarding the momentum of the particle and vice versa. This is in sharp contrast to the classical case, where both the position and momentum of the classical particle are exactly known at every instant. One might object that even in a classical measurement a certain amount of en-ergy has to be imparted to the particle being observed. However, in principle, in classical physics the energy imparted in a process of classical measurement can be made arbitrarily precise; in contrast, in quantum physics, to carry out the mea-surement with a precision of, say, x will necessarily involve at least one photon , having a minimum momentum of h= and minimum energy hc= . In other words, in quantum mechanics, the momentum and energy required to make a measure-ment of a given precision cannot be made vanishingly small. This, in essence, is the dividing line between classical and quantum measurement theory: namely, the nonvanishing value of Planck’s constant.
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