Heisenberg uncertainty relation

The energy q of a nuclear or electronic excited state of mean lifetime t cannot be determined exactly because of the limited time interval At available for the measurement. Instead, q can only be established with an inherent uncertainty, AE, which is given by the Heisenberg uncertainty relation in the form of the conjugate variables energy and time,... 

Another Heisenberg uncertainty relation exists for the energy E ofa particle and the time t at which the particle has that value for the energy. The uncertainty Am in the angular frequency of the wave packet is related to the uncertainty A in the energy of the particle by Am = h.E/h, so that the relation (1.25) when applied to a free particle becomes... 

Combining equations (1.46) and (1.47), we see that AEAt = AxAp. Thus, the relation (1.45) follows from (1.44). The Heisenberg uncertainty relation (1.45) is treated more thoroughly in Section 3.10. 

Using the results of Section 4.4, we may easily verify for the harmonic oscillator the Heisenberg uncertainty relation as discussed in Section 3.11. Specifically, we wish to show for the harmonic oscillator that... 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

In these Lorentzian lines, the parameter x describes the kinetic decay lifetime of the molecule. One says that the spectral lines have been lifetime or Heisenberg broadened by an amount proportional to l lx. The latter terminology arises because the finite lifetime of the molecular states can be viewed as producing, via the Heisenberg uncertainty relation AEAt > -h, states whose energy is "uncertain" to within an amount AE. 

In Fig. 19 the new more general uncertainty relations are represented in a solid line for three finite values of the size of the basic wavelet. The same picture also shows the plot of the usual Heisenberg uncertainty relations... 

Figure 19. Plot for the new more general uncertainty relations (solid lines) for three different values of size the basic wavelet. The usual Heisenberg uncertainty relations are also represented (dashed line).

Consider the following experiment. An electron source emits, at fixed known time t0, electrons with an uncertainty in position Ajcq 100A, which, according to Heisenberg uncertainty relations, corresponds to a minimum velocity dispersion Avo 10 6 cm. A millisecond later, the initial wavepacket will have spread to a size of about 10 m. Let us now suppose that from each electron wavepacket one cuts a piece of size 8xq 100 A (see Fig. 25). 

Until the advent of superresolution microscopes the only way to observe the minute world was based on the common Fourier-type microscopes. The maximum theoretical resolution limit for these apparatuses was established by Abbe, applying the Rayleigh-Fourier diffraction resolution rule [28]. The basic principle underlying the operational working of these ordinary microscopes is, in the eyes of Niels Bohr, a textbook example of Heisenberg uncertainty relations. 

J.R.Croca, Experimental violation of Heisenberg uncertainty relations, talkat 5thUKConf. on Conceptual and Philosophical Problems in Physics, Oxford, Sept. 1996. 

The possibly transferred rotational energy is considered as restricted according to Heisenbergs uncertainty relation (AE ft /tc > /rc) by the lifetime of the collisional intermediate... 

For liquids with rapid intermolecular motion, 7j is a measure of the life-time of a nucleus in a particular spin state. According to the Heisenberg uncertainty relation,... 

Now the modified covariant Heisenberg uncertainty relations can be written as... 

Unlike classical mechanics, a precise simultaneous specification of position and momentum is not possible. Thus, any valid state will at all times obey the Heisenberg uncertainty relation... 

Coherent States as Minimizing the Heisenberg Uncertainty Relation Statistical Equilibrium Density Operator of a Coherent State... 

N.3 COHERENT STATES AS MINIMIZING THE HEISENBERG UNCERTAINTY RELATION... 

The challenges come from Refs. [1, 7, 8, 10]. The Copenhagen view on QM requires the existence of a classical macroscopic domain in order to explain the measurement process. Heisenberg uncertainty relations appear as the mathematical expression of a complementarity concept, quantifying the mutual disturbance that takes place in a simultaneous measurement of incompatible observables, say A and 6, that is, operators that do not commute. 

Electron Distributions and Nuclear Distributions the Heisenberg Uncertainty Relation and Molecular Shape... 

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