Statement of the Heisenberg Uncertainty Principle

Heisenberg, by considering a hypothetical experiment in which the position and momentum of an electron were deduced from observations 

it is not possible to know simultaneously both the precise position and the momentum of a microscopic particle, such as an electron or atom. 

Q For a particle moving freely along the x axis, show that the Heisenberg uncertainty principle can be written in the alternative form  

A Differentiation of the de Broglie relation, p = MX, gives dp/dA = -MX. The uncertainty in momentum, Ap, can be equated with dp and the uncertainty in wavelength, AA, with -dA (the uncertainties must always be positive). Thus  

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This general expression relates the uncertainties in the simultaneous measurements of A and B to the commutator of the corresponding operators A and B and is a general statement of the Heisenberg uncertainty principle. 

The average or mean life is also of fundamental physical significance because it is the time to be substituted in the mathematical statement of the Heisenberg uncertainty principle, that is,... 

To help us understand the nature of an orbital, we need to consider a principle discovered by Werner Heisenberg, one of the primary developers of quantum mechanics. Heisenberg s mathematical analysis led him to a surprising conclusion There is a fundamental limitation to just how precisely we can know both the position and momentum of a particle at a given time. This is a statement of the Heisenberg uncertainty principle. Stated mathematically, the uncertainty principle is... 

Equation (5.13) is the quantitative statement of the Heisenberg uncertainty principle. 

Equation (5.13) is usually considered to be the quantitative statement of the Heisenberg uncertainty principle (Section 1.3). However, the meaning of the standard deviations in Eqs. (5.12) and (5.13) is rather different than the meaning of the uncertainties in Section 1.3. To find Ax in (5.13) we take a very large number of systems, each of which has the same state function and we perform one measurement of x in each system. From these measured values, symbolized by we calculate (x) and the squares of the deviations (x, - (x)). We average the squares of the deviations to get the variance and take the square root to get the standard deviation cr(x) = Ax. Then we take many systems, each of which is in the same state 4 as used to get Ax, and we do a single measurement of p c in each system, calculating Ap from these measurements. Thus, the statistical quantities Ax and Ap in (5.13) are not errors of individual measurements and are not found from simultaneous measurements of x and Px (see Ballentine, pp. 225-226). 

The breakthrough in understanding atomic structure came in 1926, when the Austrian physicist Erwin Schrodinger (1887-1961) proposed what has come to be called the quantum mechanical model of the atom. The fundamental idea behind the model is that it s best to abandon the notion of an electron as a small particle moving around the nucleus in a defined path and to concentrate instead on the electron s wavelike properties. In fact, it was shown in 1927 by Werner Heisenberg (1901-1976) that it is impossible to know precisely where an electron is and what path it follows—a statement called the Heisenberg uncertainty principle. 

The correct answer is (C). The statement in 11 is the essence of the Heisenberg Uncertainty Principle. 

Organic chemists do not think of molecules only in terms of atoms, however. We often envision molecules as collections of nuclei and electrons, and we consider the electrons to be constrained to certain regions of space (orbitals) around the nuclei. Thus, we interpret UV-vis absorption, emission, or scattering spectroscopy in terms of movement of electrons from one of these orbitals to another. These concepts resulted from the development of quantum mechanics. The Bohr model of the atom, the Heisenberg uncertainty principle, and the Schrodinger equation laid the foundation for our current ways of thinking about chemistry. There may be some truth in the statement that... 

Heisenberg uncertainty principle - The statement that two observable properties of a system that are complementary, in the sense that their quantum-mechanical operators do not commute, cannot be specified simultaneously with absolute precision. An example is the position and momentum of a particle according to this principle, the uncertainties in position Aq and momentum Ap must satisfy the relation ApAq > /z/4tt, where h is Planck s constant. 

Specifically, a statement known as the principle of uncertainty - or Heisenbergs Uncertainty Principle - tells us that once you know the interaction energy of two atoms in a molecule you cannot know their positions very accurately. This relation, named in honor of the German physicist Werner Heisenberg, can be expressed in the following way ... 

The generally accepted notion of wave-particle duality, which predates Heisenberg and Bohr, could be reconciled with the probability interpretation, but the fuzziness associated with waves remained unexplained in the orthodox tradition. The proclamation of the quantum-mechanical uncertainty principle was intended to take care of the oversight. A more serious indictment of the orthodox tradition is hard to imagine, short of the blunt statement by Nobel physicist, Murray Gell-Mann [28] ... 

As we noted at the end of Chapter 2, the success of Bohr s theory was short-lived. Emission spectra of multi-electron atoms (recall that the hydrogen atom has only one electron) could not be explained by Bohr s theory. DeBroglie s statement that electrons have wave properties served to intensify the problem. Bohr stated that electrons in atoms had very specific locations. The very nature of waves, spread out in space, defies such an exact model of electrons in atoms. Furthermore, the exact model is contradictory to Heisenberg s Uncertainty Principle. 

If an electron has wave-like properties, there is an important and difficult consequence it becomes impossible to know exactly both the momentum and position of the electron at the same instant in time. This is a statement of Heisenberg s uncertainty principle. In order to get around this problem, rather than trying to define its exact position and momentum, we use the probability of finding the electron in a given volume of space. The probability of finding an electron at a given point in space is determined from the function ij where is a mathematical function which describes the behaviour of an electron-wave ip is the wavefunction. 

Perhaps the most unusual part of quantum mechanics is the statement called the uncertainty principle. Occasionally it is called Heisenberg s uncertainty principle or the Heisenberg principle, after the German scientist Werner Heisenberg (Figure 10.2), who announced it in 1927. The uncertainty principle states that there are ultimate limits to how exact certain measurements can be. This idea was problematic for many scientists at the time, because science itself was concerned with finding specific answers to various questions. Scientists found that there were limits to how specific those answers could be. 

With some algebra, it follows from Eq. (2.65) that the product of the uncertainties (root-mean-square deviations) in the expectation values for position and momentum must be > Hjl [4]. This is a statement of Heisenberg s uncertainty principle. 

We have seen that, according to the de Broglie relation, a wave of constant wavelength, the wavefunction sin(2nx/X), corresponds to a particle with a definite linear momentum p = h/L However, a wave does not have a definite location at a single point in space, so we cannot speak of the precise position of the particle if it has a definite momentum. Indeed, because a sine wave spreads throughout the whole of space, we cannot say anything about the location of the particle because the wave spreads everywhere, the particle maybe found anywhere in the whole of space. This statement is one half of the uncertainty principle, proposed by Werner Heisenberg in 1927, in one of the most celebrated results of quantum mechanics ... 

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