Heisenberg uncertainity principle

Heisenberg uncertainly principle, as an expectation value. Streitwieser has eloquentiy summarized the importance of such concepts in chemical computations (42) ... 

The quantity 4 2 represents the probability of finding the electron in the described region. It is also interpreted statistically, in the context of the Heisenberg uncertainly principle, as an expectation value. Streitwieser has eloquently summarized the importance of such concepts in chemical computations (42) ... 

The Heisenberg Uncertainly Principle arises from the dual nature (wave-particle) of matter. It states that there exists an inherent uncertainty in the product of the position of a particle and its momentum, and that this uncertainty is on the order of Planck s constant. 

An important feature of operators is that they may not commute, i.e. for two particular operators A and B one may have AB — BA 0. This property has important physical consequences (see below, postulate IV and the Heisenberg uncertain principle). Because of the possible non-commutation of the operators, transformation of the classical formula (in which the commutation or non-commutation did not matter) may be non-unique. In such a case, from all the possibilities one has to choose an operator which is Hermitian. The operator A is Hermitian if, for any functions ip and < > liom its domain, one has commutalion... 

In the beginning of the twentieth century, some scientists thought that a nucleus may contain both electrons and protons. U.se the Heisenberg uncertainly principle to show that an electron cannot be confined within a nucleus. Repeat the calculation for a proton. Comment on your results. A.ssume the radius of a nucleus to... 

Linewidth is governed by the Heisenberg uncertainty principle, which says that the shorter the lifetime of the excited state, the more uncertain is its energy ... 

E.g. sharp momentum and position states. At least one of such a conjugate pair is dictated by Heisenberg s principle to be statistically uncertain. 

A German physicist, Werner Heisenberg, formalized this idea with his uncertainly principle, which says that we cannot know the exact location or motion of an electron. Although we cannot describe the path of an electron s motion, we can speak of the probability of finding it in a given location at any time. In Figure 8.7, the darker areas represent a greater probability that the electron is present in this location. 

Quantum theory dictates that the measurement of certain pairs of properties of particles, including position and momentum are limited by the Heisenberg uncertainty principle first advanced by German physicist Werner Heisenberg (1901-1976). In essence, although it is possible to measure either position or momentum the pair can not be measured simultaneously. The more exact the determination of position, the more uncertain becomes the measurement of momentum. 

Thinking Critically Use de Broglie s wave-particle duality and the Heisenberg uncertainty principle to explain why the location of an electron in an atom is uncertain. 

Footnote The Wave Nature of the Electron. So far the electron has been considered as a particle, with clearly quantised energy levels, that can be precisely measured, as in the emission lines of the spectrum of hydrogen. Because the electron is so small and light, the accuracy with which it can be measured is very uncertain. This is associated with the Heisenberg Uncertainty Principle, which states that it is impossible to determine both the position and momentum of an electron simultaneously , i.e. Ax Ap = hl2it, where Ax is the uncertainty in measuring the position of the electron and Ap is the uncertainty in measuring the momentum (p = mass X velocity) of the electron. The two uncertainties bear an inverse relationship to each other. Consequently, if the position of the... 

Equation 6-26 is one of several ways of formulating the Heisenberg uncertainty principle. The meaning in words of this equation is as follows. If the energy of a particle or system of particles — photons, electrons, neutrons, or protons, for example — is measured for an exactly known period of time Ar, then this energy is uncertain by at least h t. Therefore, the energy of a particle can be known with zero uncertainty only if it is observed for an infinite period. For finite periods, the energy measurement can never be more precise than fi/Ar. The practical consequences of this limitation will appear in several of the chapters that follow. 

These oscillations define the particle s de Broglie wavelength, and therefore its momentum. If we wanted to know the momentum exactly, we would need a perfect sine wave that extended forever consequently we would not know anything about its position. If we wanted to know its position exactly, the wavepacket would become infinitely narrow along the x axis and the wavelength (and hence momentum) would become an unknowable parameter. As the position of a quantum particle wavepacket becomes more certain, its momentum becomes more uncertain, and vice versa. We cannot know exact values of both the position and momentum of the particle at the same time. That is the qualitative version of the Heisenberg imcertainty principle. More precisely, the uncertainty... 

We use Heisenberg s uncertainty principle with the equality sign to calculate the miniimim uncertainly. 

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