Classical uncertainty

Now D is inversely proportional to a macroscopic (transport) cross section which itself is proportional to Avogadro s number N. Hence, D and the product of indeterminacy is proportional to 1 /N. This feature is characteristic of classical uncertainty relations. N molecules form, so to speak, a critical domain. 

We illustrate the basic restriction of such methods with an example. The Fourier transform of a sine wave gives the delta function. To obtain the delta function, in fact, we must integrate over a time interval from —oo to +oo. If over finite time interval a wave train is integrated, then in the Fourier transform contributions of other frequencies creep in that are not there originally in fact. This is known as the classical uncertainty relation. 

In Figure 5c, we observe the same time-domain cosine wave as in Figure 5a, but for only a finite period, T sec. The result is that the frequency spectrum is now broadened from an infinitely sharp spike to a signal whose frequency width is of the order of (1/T) Hz. This result is an example of a classical "uncertainty principle" the product of the time-domain width (T) and the frequency-domain width (1/T) is constant. In other words, the only way to determine the frequency of a time-domain signal with perfect accuracy (i.e., infinite frequency "resolution") is to observe it for an infinite length of time. 

From this expression, it is seen that the temporal properties of the incident radiation field depend upon the coherence time Atg and the pulse duration AT. It must be pointed out that the physical nature of these times is essentially different. For a given source, At is imposed by the uncertainty principle of quantum theory, whereas AT corresponds to a classical statistical probability distribution, which can in principle be modified by the experimenter. In particular, if one reduces this classical uncertainty" sufficiently, the time resolution conditions can be improved so as to have the lowest possible resolution time given by the quantal uncertainty" value At. This corresponds to the minimum time uncertainty situation, for which one has S(fo) (to) and therefore p to) a(to)-The spectral properties of the incident beam are best described by... 

The first type uses irradiation by coherent monochromatic light. It corresponds to the Fourier-transform-limited situation considered in Section II,C,1, for which no classical uncertainty needs to be taken into account and for which the coherence time At is very long. For the incident beam, one has simply... 

Long pulse excitation, where At > h/5e. Here, the time interference effects are generally not detected, either because they are erased by the classical uncertainty AT (if AT > or because no initial coherence was prepared (if At, > hjde ). In any case, a necessary condition for detecting at least the monotonic decay of the excited level populations is At [Pg.320]

For each degree of freedom, classical states within a small volume A/ij Aq- h merge into a single quantum state which cannot be fiirther distinguished on account of the uncertainty principle. For a system with /... 

Many of the earlier uncertainties arose from apparent disagreements between the theoretical values and experimental detemiinations of the critical exponents. These were resolved in part by better calculations, but mainly by measurements closer and closer to the critical point. The analysis of earlier measurements assumed incorrectly that the measurements were close enough. (Van der Waals and van Laar were right that one needed to get closer to the critical point, but were wrong in expectmg that the classical exponents would then appear.) As was shown in section A2.5.6.7. there are additional contributions from extended scaling. 

Table A2.5.1 shows the Ising exponents for two and tluee dimensions, as well as the classical exponents. The uncertainties are those reported by Guida and Ziim-Justin [27]. These exponent values satisfy the equalities (as they must, considering the scalmg assumption) which are here reprised as functions of p and y ...

The small uncertainties in the calculated exponents seem to preclude the possibility that the d = 3 exponents are rational numbers (i.e. the ratio of integers). (At an earlier stage this possibility had been suggested, since not only die classical exponents, bnt also tlie rational numbers pre-RG calculations had suggested p = 5/16 and y = 5/4.)... 

Approximation Property We assume that the classical wavefunction 4> is an approximate 5-function, i.e., for all times t G [0, T] the probability density 4> t) = 4> q,t) is concentrated near a location q t) with width, i.e., position uncertainty, 6 t). Then, the quality of the TDSCF approximation can be characterized as follows ... 

The electromagnetic spectrum is a quantum effect and the width of a spectral feature is traceable to the Heisenberg uncertainty principle. The mechanical spectrum is a classical resonance effect and the width of a feature indicates a range of closely related r values for the model elements. 

Of the variety of quantum effects which are present at low temperatures we focus here mainly on delocalization effects due to the position-momentum uncertainty principle. Compared to purely classical systems, the quantum delocalization introduces fluctuations in addition to the thermal fluctuations. This may result in a decrease of phase transition temperatures as compared to a purely classical system under otherwise unchanged conditions. The ground state order may decrease as well. From the experimental point of view it is rather difficult to extract the amount of quantumness of the system. The delocahzation can become so pronounced that certain phases are stable in contrast to the case in classical systems. We analyze these effects in Sec. V, in particular the phase transitions in adsorbed N2, H2 and D2 layers. 

Traceability of measurement results is essential in the establishment of a certified reference material. As stipulated in ISO Guides 30 and 35, a certified reference material can only be certified if there is an uncertainty statement with a traceability statement. Basically, traceability means anchoring. In classical analytical chemistry, that SI system is often the best choice as a reference (= anchoring poinf). However, there is a wide range of parameters either defined by a method or defined by the... 

The Heisenberg uncertainty principle is a consequence of the stipulation that a quantum particle is a wave packet. The mathematical construction of a wave packet from plane waves of varying wave numbers dictates the relation (1.44). It is not the situation that while the position and the momentum of the particle are well-defined, they cannot be measured simultaneously to any desired degree of accuracy. The position and momentum are, in fact, not simultaneously precisely defined. The more precisely one is defined, the less precisely is the other, in accordance with equation (1.44). This situation is in contrast to classical-mechanical behavior, where both the position and the momentum can, in principle, be specified simultaneously as precisely as one wishes. 

Werner Heisenberg (1901-1976 Nobel Prize for physics 1932) developed quantum mechanics, which allowed an accurate description of the atom. Together with his teacher and friend Niels Bohr, he elaborated the consequences in the "Copenhagen Interpretation" — a new world view. He found that the classical laws of physics are not valid at the atomic level. Coincidence and probability replaced cause and effect. According to the Heisenberg Uncertainty Principle, the location and momentum of atomic particles cannot be determined simultaneously. If the value of one is measured, the other is necessarily changed. 

In classical mechanics both the position of a particle and its velocity at any given instant can be determined with as much accuracy as the experimental procedure allows. However, in 1927 Heisenberg introduced the idea that the wave nature of matter sets limits to the accuracy with which these properties can be measured simultaneously for a very small particle such as an electron. He showed that Ax, the product of the uncertainty in the measurement of the position x, and Ap, the uncertainty in the measurement of the momentum p, can never be smaller than M2tt ... 

In principle, all measurements are subject to random scattering. Additionally measurements can be affected by systematic deviations. Therefore, the uncertainty of each measurement and measured result has to be evaluated with regard to the aim of the analytical investigation. The uncertainty of a final analytical result is composed of the uncertainties of all the steps of the analytical process and is expressed either in the way of classical statistics by the addition of variances... 

Traditionally, analytical chemists and physicists have treated uncertainties of measurements in slightly different ways. Whereas chemists have oriented towards classical error theory and used their statistics (Kaiser [ 1936] Kaiser and Specker [1956]), physicists commonly use empirical uncertainties (from knowledge and experience) which are consequently added according to the law of error propagation. Both ways are combined in the modern uncertainty concept. Uncertainty of measurement is defined as Parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand (ISO 3534-1 [1993] EURACHEM [1995]). 

The uncertainty concept is composed of both chemists and physicists approaches of handling of random deviations and substitutes so classical error theories in an advantageous way. 

According to classical information theory, founded by Shannon [1948] (see also Shannon and Weaver [1949]), information is eliminated uncertainty about an occurrence or an object, obtained by a message or an experiment. Information is always bound up with signals. They are the carriers of information in the form of definite states or processes of material systems (Eckschlager and Danzer [1994] Danzer [2004]) see Sect. 3.1. 

The connection between the multiplicative insensitivity of 12 and thermodynamics is actually rather intuitive classically, we are normally only concerned with entropy differences, not absolute entropy values. Along these lines, if we examine Boltzmann s equation, S = kB In 12, where kB is the Boltzmann constant, we see that a multiplicative uncertainty in the density of states translates to an additive uncertainty in the entropy. From a simulation perspective, this implies that we need not converge to an absolute density of states. Typically, however, one implements a heuristic rule which defines the minimum value of the working density of states to be one. 

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