Heisenberg Uncertainty Principle

12 Key excerpts on "Heisenberg Uncertainty Principle"

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  Exploring Quantum Physics through Hands-on Projects

  • David Prutchi(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  symbol means in other books and papers on quantum physics.

  Heisenberg’s Uncertainty Principle was published in a 1927 paper that argued why the position and momentum of a particle cannot both be measured exactly, at the same time, even in theory. For Heisenberg, and later for the whole physics society centered on Bohr’s Copenhagen school, the very concepts of exact position and exact momentum together, in fact, had no meaning in nature.

  Please note that the uncertainty in the measurement of either position or momentum is not due to lack of accuracy of our measurement instruments. The compromise in the measurements happens in principle, as a law of nature. That is, even if your measurement instruments are infinitely precise, you cannot simultaneously know the position and momentum of a particle with absolute precision. For this reason, some people prefer to call this concept the Indeterminacy Principle, so that the intrinsic indeterminate character of nature is implied.

  Notice, however, that the limitation is in the order of magnitude of Planck’s constant h. In fact, since we are dividing it by 4π, it is one order of magnitude lower than Planck’s constant. This means that the Uncertainty Principle doesn’t really affect our everyday experience, in which momentum is so much larger than Planck’s constant.

  EXPERIMENTAL DEMONSTRATION OF THE UNCERTAINTY PRINCIPLE

  A demonstration of the Uncertainty Principle is simple to conduct using a laser pointer and an adjustable slit consisting of two razor blades. The idea is that the photons emitted by the laser travel without diverging. As shown in Figure 104 , the photons exiting the laser can be assumed to have momentum only along the x axis (so py

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  Physical Chemistry

  How Chemistry Works

  • Kurt W. Kolasinski(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  However, there are certain questions regarding incompatible observables about which our knowledge is limited. A much better name would be the ‘indeterminacy principle.’ The Heisenberg relations and the Schrödinger equation define which questions can be determined exactly and which have approximate answers. The uncertainty principle is not an assumption. Rather, it is a direct consequence of the postulates of quantum mechanics, in particular the statistical interpretation inherent in the Born rule and the definition of expectation values. The most general form of the uncertainty principle is written in terms of the expectation value of the commutator (20.85) for any pair of operators whose commutator comprises an operator of the form (20.86) Is the strange behavior observed for an electron (that the measurement perturbs the object of measurement) but not for a snooker ball (or other macroscopic object) a property of measurement or a property of the particle itself? We have already encountered this question in Section 19.5.1. The answer in quantum mechanics is both. Measurement interactions affect the particle and the accuracy of the measurement – even for macroscopic particles such as the roughly 10 kg objects that are to be used to measure gravitation waves 19. However, the collapse of the wavefunction by the interaction of the particle with the environment is also a fundamental property of the particle and the basis of one of the postulates of quantum mechanics 20. 20.10.3 Position–momentum uncertainty relationship Assume that we have a particle in a state described by the wavefunction (20.87) where a is a normalization constant and 0 ≤ x ≤ ∞. Let us determine the expectation value of the commutator of the position and momentum operators so as to derive their uncertainty relation. The position and momentum operators are and

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  How to Teach Quantum Physics to Your Dog

  • Chad Orzel(Author)
  • 2009(Publication Date)
  • Scribner

    (Publisher)

  She trots over to the TV, and sticks her nose under the cabinet. “Oooh! 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.

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  How to Teach Quantum Physics to Your Dog

  • 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

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  Quantum Mechanics

  Its Early Development and the Road to Entanglement and Beyond

  • Edward G Steward(Author)
  • 2011(Publication Date)
  • ICP

    (Publisher)

  §9.1 .

  It is necessary to emphasise that the Uncertainty Principle is not merely a result of practical limitations in making measurements of, say, p and q. It is only when one property is observed/measured that that property becomes definite, and similarly for the other: unobserved, the particle has no definite p or q.

  A simple way (Bohr 1927, 1928) of approaching the Uncertainty Principle, on the wave–particle basis, is to consider a wavepacket representing a mass particle. Let the wave-packet have length Δq. This then represents the uncertainty in determining the exact position of the associated particle. Its length is proportional to the associated wavelength range, Δλ say, comprising the wave-packet, and from the Bandwidth Theorem, we have

  Now Δλ also relates to the uncertainty in evaluating the momentum of the particle, since de Broglie’s Eqn. 7.16 tells us that

  Therefore, Thus, like Bohr, we arrive at Eqn. 9.3,

  In his paper, Heisenberg went on to consider other things such as the path, velocity, and energy of an electron. For example, he considered the concept of an electron orbit in an atom. By ‘electron orbit’ he said, “we mean a series of points at which the electron occupies one after another as ‘position’ ”. However, illumination of a sufficiently short wavelength to ‘measure’ the orbit would result in the ejection of the electron from its orbit: any sense of orbit would be lost in the process of observing it. The only way of ‘detecting’ the whole orbit would be to envisage making repeated measurements on a single atom — or, more realistically, measurements on an ensemble of the atoms. This would amount to a probability distribution function for the position of the electron, and here Heisenberg was acknowledging its relationship to Born’s probability interpretation of the Schrödinger wave function — adding to Heisenberg’s alienation from Bohr. At each ‘measurement’ there is what became known as a ‘collapse’ of the wave function Ψ (cf. §7.3.9

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  Structure of Space and the Submicroscopic Deterministic Concept of Physics

  • 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 2

  ħ

  based 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

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  Quantum Mechanics I

  The Fundamentals

  • S. Rajasekar, R. Velusamy(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  5.3 ×

  10

  − 11

    m. 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

  5.3 ×

  10

  − 11

   

  m.
• 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.
• Suppose

  ψ ( x )

  is a delta-function at

  x =

  x 0

  . The position of the particle is determined exactly. What about the corresponding momentum wave function? The momentum wave function

  C ( k )

  is a sinusoidal function extending over all space. Thus, the momentum uncertainty is infinity. Similarly, if

  ψ ( 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