Uncertainty-principle considerations

The continuity of tunneling and ballistic transport for atom-scaled potential barriers is a straightforward consequence of the uncertainty principle. It can be shown from the following two different points of view  

The first view is based on the uncertainty relation between time and energy. When the electron is in the region of the barrier, its velocity, as determined by the kinetic energy, is  

The energy uncertainty for an electron in the barrier region is then 

Ai E — Ub = 2 eV and W = 3 % A, the energy uncertainty is AE 2.5 eV. In other words, the energy uncertainty is larger than the absolute value of the kinetic energy. Therefore, for atom-scale barriers, the distinction between tunneling and ballistic transport disappears. 

The second view is based on the uncertainty relation between coordinate and momentum. In the region of a classically allowed barrier, the de Broglie wavelength of an electron is 

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For FTS data, artifact removal is a consideration that is as important as resolution improvement for most researchers in this field. Interferogram continuation methods are not as yet widely known in this area. Methods currently in widespread use that are effective in artifact removal involve the multiplication of the interferogram by various window functions, an operation called apodization. A carefully chosen window function can be very effective in suppressing the artifacts. However, the peaks are almost always broadened in the process. This can be understood from the uncertainty principle. A window that reduces the function most strongly closest to the end points will yield a transform for the modified function that must be broader than it was originally. Alternatively we may employ the convolution... 

In the consideration of the momentum of a large number of particles restricted to a volume V, it is often convenient to describe the system by an assembly of points in a momentum diagram (Fig. 1). The length OA represents the magnitude of momentum of the particle A, and its direction is OA. The application of Heisenberg s uncertainty principle leads to the... 

If you look at the nmr spectra of many different kinds of organic compounds, you will notice that some resonances are sharp and others are broad. In a few spectra, all of the peaks may be broad as the result of poor spectrometer performance, but this is not true for the spectra of Figures 9-29 (p. 312) and 24-2 (p. 1173) where, within a given spectrum, some resonances will be seen to be sharp and others broad. We can understand these differences by consideration of the lifetimes of the magnetic states between which the nmr transitions occur.1 The lifetimes of the states can be related to the width of the lines by the Heisenberg uncertainty principle. 

A main feature of ultrafast processes under consideration takes place in the time scale shorter than picoseconds. Thus, it is necessary to employ the laser with pulse-duration 10 fsec to study these ultrafast processes. From the uncertainty principle AE At h/2 it can be seen that using this pulse-duration, numerous vibronic states can be coherently pumped (or excited) and thus the probing signal in a pump-probe experiment will contain the information of the dynamics of both population and coherence (or phase). In other words, in order to obtain the information of ultrafast dynamics it is... 

While h is quite small in the macroscopic world, it is not at all insignificant when the particle under consideration is of subatomic scale. Let us use an actual example to illustrate this point. Suppose the Ax of an electron is 10-14 m, or 0.01 pm. Then, with eq. (1.2.1), we get Apx = 5.27 x 10-21 kg m s-1. This uncertainty in momentum would be quite small in the macroscopic world. However, for subatomic particles such as an electron, with mass of 9.11 x 10-31 kg, such an uncertainty would not be negligible at all. Hence, on the basis of the Uncertainty Principle, we can no longer say that an electron is precisely located at this point with an exactly known velocity. It should be stressed that the uncertainties we are discussing here have nothing to do with the imperfection of the measuring instruments. Rather, they are inherent indeterminacies. If we recall the Bohr theory of the hydrogen atom, we find that both the radius of the orbit and the velocity of the electron can be precisely calculated. Hence the Bohr results violate the Uncertainty Principle. 

The name zero-point energy is used for the energy of a system in its lowest stationary state because the system in thermodynamic equilibrium with its environment at a temperature approaching the absolute zero would be in this stationary state. The zero-point energy is of considerable importance in many statistical-mechanical and thermodynamic discussions. The existence of zero-point energy is correlated with the uncertainty principle (Chap. XV),... 

The branch of quantum mechanics to which we have devoted our attention in the preceding chapters, based on the Schrodinger wave equation, can be applied in the discussion of most questions which arise in physics and chemistry. It is sometimes convenient, however, to use somewhat different mathematical methods and, moreover, it has been found that a thoroughly satisfactory general theory of quantum mechanics and its physical interpretation require that a considerable extension of the simple theory be made. In the following sections we shall give a brief discussion of matrix mechanics (Sec. 51), the properties of angular momentum (Sec. 52), the uncertainty principle (Sec. 53), and transformation theory (Sec. 54). 

For most gas molecules, there are several hundreds, even thousands, of possible molecular energy states. Therefore, one expects to find a large number of wavelengths at which the molecules absorb the incident energy, which makes the prediction of gas absorption a very difficult problem. Additionally, each of these absorption wavelengths can be broadened because of pressure and temperature as well as the uncertainty principle of Heisenberg. It is obvious that exact consideration of all these active frequencies/wavelengths may not be... 

An important consideration in spectroscopic measurements concerns the bandwidth of the laser sources. In order to resolve the vibrational resonances in a conventional approach, one needs, in the conventional scheme, a tunable source that has a narrow bandwidth compared to the resonance being studied. For t5q)ical resolutions, this requirement implies, by uncertainty principle, that IR pulses of picosecond or longer duration must be used longer. On the other hand, ultrafast pulsed IR sources with broad bandwidths are quite attractive from the experimental standpoint. In order to make use of these sources, two t5q)es of new experimental techniques have been introduced. One technique involves mixing the broadband IR source ( 300 cm ) with a narrowband visible input ( 5 cm ). By spectrally resolving the SF output, we may then obtain resolution of the IR spectrum limited only by the linewidth of the visible source [M, M]- This result follows from the fact that SF vis satisfied for the SFG process. The second new approach involves the... 

The accuracy of the average risk of a classification can be computed in principle if the statistics of the data are known. However, the uncertainties mentioned considerably limit the computation of risk in practical applications. Therefore, the usefulness of a parametric method for a practical classification problem can be examined - as for other classification methods - only empirically. 

The main consequence of the uncertainty principle is that, because electronic energy levels are known with considerable accuracy, the positions of electrons within atoms are not known at all accurately. This realization forces theoretical chemistry to develop methods of calculation of electronic positions in terms of probabilities rather than assigning to them, for example, fixed radii around the nucleus. The varied methods of these calculations are known collectively as quantum mechanics. 

If, however, the two particles are farther apart than 1.4 fm, as in B, the emitted messenger particle is not able to traverse the distance between the particles, but instead turns back and disappears. There is accordingly no interaction between the nucleons at the larger distance. The reason that the range of the messenger particles is restricted can be understood by consideration of the uncertainty principle. 

The examples of phase transitions mentioned above occur at nonzero temperature. At these so-called thermal or classical transitions, the ordered phase (the ice crystal or the ferromagnetic state of iron) is destroyed by thermal fluctuations. In the last two decades or so, considerable attention has focused on a very different class of phase transitions. These new transitions occur at zero temperature when a nonthermal parameter such as pressure, chemical composition, or magnetic field is changed. The fluctuations that destroy the ordered phase in these transitions cannot be of a thermal nature. Instead, they are quantum fluctuations that are a consequence of Heisenberg s uncertainty principle. For this reason, these zero-temperature transitions are called quantum phase transitions. 

As a result of these short lifetimes, electron attachment energies for anion shape resonances are considerably broadened by the time-energy uncertainty principle. 483 p j. lifetime At, the resonance energy is subject to broadening according to ... 

Scale-Up Principles. Key factors affecting scale-up of reactor performance are nature of reaction zones, specific reaction rates, and mass- and heat-transport rates to and from reaction sites. Where considerable uncertainties exist or large quantities of products are needed for market evaluations, intermediate-sized demonstration units between pilot and industrial plants are usehil. Matching overall fluid flow characteristics within the reactor might determine the operative criteria. Ideally, the smaller reactor acts as a volume segment of the larger one. Elow distributions are not markedly influenced by... 

The considerations presented above were based on the specific assumption that the catalytic reaction of the serine proteases involves mechanism a of Fig. 7.2. However, one can argue that the relevant mechanism is mechanism b (the so-called charge-relay mechanism ). In principle the proper procedure, in case of uncertainty about the actual mechanism, is to perform the calculations for the different alternative mechanisms and to find out which of the calculated activation barriers reproduces the observed one. This procedure, however, can be used with confidence only if the calculations are sufficiently reliable. Fortunately, in many cases one can judge the feasibility of different mechanisms without fully quantitative calculations by a simple conceptual consideration based on the EVB philosophy. To see this point let us consider the feasibility of the charge-relay mechanism (mechanism b) as an alternative to mechanism a. Starting from Fig. 7.2 we note that the energetics of route b can be obtained from the difference between the activation barriers of route b and route a by... 

As in the previous chapter, most work has been carried out on oxides, and these figure prominently here. As the literature on oxides alone is not only vast but is also rapidly increasing, this chapter focuses upon a number of representative structure types to explain the broad principles upon which the defect chemistry depends. However, despite considerable research, the defect chemistry and physics of doped crystals is still open to considerable uncertainty, and even well-investigated simple oxides such as lithium-doped nickel oxide, Li Nij- O, appear to have more complex defect structures than thought some years ago. 

The introductory Chapter 2 of the Air Quality Guidelines (WHO 2000) gives a very detailed and comprehensive description of the criteria used in establishing the guideline values including criteria for selection of NOAEL/LOAEL, adverse effect, benchmark approach, and uncertainty factors. These criteria are comparable to the principles outlined in Chapters 4 and 5 in this book. There are also criteria for selection of averaging times and for consideration of sensory effects (malodorous... 

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