Applications of the Uncertainty Principle

The Uncertainty Principle often permits simple explanations of complex quantum mechanical results. Here are a few examples. 

Confining a particle to a restricted region in space increases its minimum possible energy. Let us consider one particularly simple case—a particle in a one dimensional box . The box is defined by a potential 

This potential is the same as a flat-bottomed container with infinitely high walls separating inside from outside. Here we will use the Uncertainty Principle to estimate the minimum energy later (in Chapter 6) we will find all of the possible energies and states for this system, using a differential equation known as 

The lowest energy state decreases in energy as the box expands (enlarging the box permits a larger value of Ax, hence a smaller value of Ap). Thus, for example, a ball in a box must always be moving, but it can have a lower minimum speed if the box is big. 

We could get the same answer in a different way, using de Broglie s relation A, = h/p (Problem 5-15). The wave representing the electron would have to vanish at the two walls, similar to the waves on a violin string. The longest possible wave we could fit into the box would have wavelength k = 2L. Such a wave would go through half a cycle between the two walls, and would be zero at each wall. 

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These experimental studies in mixed crystals, crystals, and in low-pressure vapors provide compelling indications that electronic relaxation and spectral diffuseness are related phenomena, and the implications are that electronic relaxation rates are directly obtainable by application of the uncertainty principle. 

This is exactly the width which is predicted by the application of the uncertainty principle ... 

DilP recently discussed the merits and limitations of models that assume thermodynamic additivity and independence (of energy types, of neighbor interactions, of conformational freedom, of monomer contact pairing frequencies, etc.). He states that biological molecules may achieve stability in the face of thermal uncertainty, as polymers do, by compounding many small interactions this summing can stump modelers because application of the additivity principle leads to accumulated error. Entropies and free energy may not be additive to describe weak interactions that are ensembles of states. He concludes that additivity principles appear to be few and limited in scope in biochemistry. 

Many questions and problems about the application of the GLP Principles centre around, and originate from, an uncertainty about the real areas of applicability. Apart from the clear-cut, classical area of application, i.e. in toxicology studies on chemicals, pesticides and pharmaceutical ingredients, there may be investigations, where it could become difficult to judge whether or not these should be conducted under the strict regime of the GLP standards. There are, however, two firm determinants which can be used to define the necessity for application of the principles of GLP. They have to be applied in combination in order to determine whether or not for any specific study or study type strict adherence to GLP would be mandatory. 

The naive application of the variational principle to the optimization problem is limited by the statistical uncertainty inherent in every Monte Carlo calculation. The magnitude of this statistical error has a great impact on the convergence of the optimization, and on the ability to find the optimal parameter set as well. 

Application of the above principles and solutions in practice have come across with a problem of uncertainty with respect to achievable efficiency the latter is connected with necessity of quantification of the safety level. To evaluate and assess the achievable safety level, one has to perform the full-scale PSA in addition to deterministic principles. Common realisation of the deterministic and probabilistic analyses during the NVAES-2 design process has permitted to obtain optimal solution with respect to balance of the active and passive trains of the modular safety systems. As a result of this optimisation, total core melt frequency for NVAES-2 is about three orders of magnitude less than for unit 1 of Balakovo NPP with V-320 reactor plant. 

The use of wave groups or wave packets in physics, and certainly in chemistry, was limited to a few theoretical examples in the applications of quantum mechanics. The solution of the time-dependent Schrodinger equation for a particle in a box, or for a harmonic oscillator, and the elucidation of the uncertainty principle by superposition of waves are two of these examples. However, essentially all theoretical problems are presented as solutions in the time-independent frame picture. In part, this practice is due to the desire to start from a quantum-state description. But, more importantly, it was due to the lack of experimental ability to synthesize wave packets. 

In many cases there may be uncertainty over what value to use for a given conditional modifier because the factors whioh influence it cannot all be defined or characterised, eg where the role of human behaviour is unoertain or where the underlying science is itself uncertain. Under these circumstances a conservative approach should be taken, consistent with the application of the precautionary principle (see paragraphs 23-24 of this appendix). 

Interest in Stark manipulation of molecular beams had been rekindled in the 1990s by the growth of cold molecules as a research topic in atomic, molecular and optical physics. Indeed, Stark deceleration has largely shaped the field of cold molecule research, as it became almost instantly the workhorse of the field. Moreover, the quantum-state selected molecular beam that exits a Stark decelerator has a tunable velocity, which is ideally suited for many applications. For instance, decelerated beams can be used in high-resolution spectroscopic studies to extend the available observation time and, by virtue of the uncertainty principle, to improve the attainable energy resolution. Decelerated beams also enable the study of (in)elastic collisions and reactive scattering down to zero collision... 

For criteria development we used systematical application of an insignificancy principle (IP). Uncertainty is insignificant in comparison with on the level of 95% if their pooled uncertainty is at most 5% exceeds A. It results in an inequality A < 0.32-A that is the main instiximent in development of acceptability criteria of validation characteristics. 

The initial set of experiments and the first few textbook chapters lay down a foundation for the course. The elements of scientific activity are immediately displayed, including the role of uncertainty. The atomic theory, the nature of matter in its various phases, and the mole concept are developed. Then an extended section of the course is devoted to the extraction of important chemical principles from relevant laboratory experience. The principles considered include energy, rate and equilibrium characteristics of chemical reactions, chemical periodicity, and chemical bonding in gases, liquids, and solids. The course concludes with several chapters of descriptive chemistry in which the applicability and worth of the chemical principles developed earlier are seen again and again. 

Discussions about the PA usually focus on definitions. Such definitions are plentiful, they depend on the scientific and social background of their authors, and they all contain elements of truth and error. One of the basic problems with the PA is that there is no such thing as an overall definition. The application of the PA is always heavily context-dependent. It is no use solving problems associated with applying the PA by means of a generally accepted definition, since it is difficult to define a principle sharply where uncertainty is the main element. The definition of terms and concepts like uncertainty always depend on the scientific, social, cultural and economic background of individuals employing them. 

The uncertainty principle will be applied in this section to nmr spectroscopy but, as we will see later, it is applicable to all other forms of spectroscopy. 

One of the features of transition state theory is that in principle it permits the calculation of absolute reaction rate constants and therefore the thermodynamic parameters of activation. There have been few successful applications of the theory to actual reactions, however, and agreement with experiment has not always been satisfactory. The source of difficulty is apparent when one realizes that there really is no way of observing any of the properties of the activated complex, for by definition its lifetime is of the order of a molecular vibration, or 10-14 sec. While estimates of the required properties can often be made with some confidence, there remains the uncertainty due to lack of independent information. 

The following discussion centers on uncertainty related to the estimation of human exposure. Many of the general principles will be applicable to both exposure and toxicology, and the connection and context of this treatment of uncertainty associated with toxicological determinations will be made later in this discussion. 

Gomer R. Field Emission and Field Ionization. Cambridge, Mass Harvard University Press 1961, pp. 6-11, 66-70, and 181-183. The application of the position-momentum uncertainty principle per se is discussed on pp. 6-8. 

The physical properties determined using the ECD are important to different areas of chemistry. Analytical chemistry deals with how much and what are involved in a chemical reaction. Expressed differently, it establishes what we refer to as the QQQ quantitation, qualitative identification, and the quality of the results. The determination of the electron affinities of the chlorinated biphenyls, dioxins, and phenols and the prediction of the response of the ECD and NIMS are important to qualitative and quantitative analyses of environmental pollutants [21]. Polarographic reduction in solutions likewise gives accurate and precise qualitative and quantitative results. The quality of the analyses is expressed by the random and systematic uncertainties in the reported values. These are obtained from the same principle of weighted least squares used to obtain information from ECD data. Wentworth has described the application of the general least-squares procedure to chemical problems [22, 23]. 

Evaluation and interpretation of the DNT and comparative cholinesterase study data are conducted in accordance with agency risk assessment guidelines for developmental toxicity, reproductive toxicity, and neurotoxicity (EPA, 1991b, 1996, 1998e). Risk as.se.ssinents. including determination of reference values and application of traditional uncertainty factors in the calculations, are conducted ba.sed on historically established principles (NAS, 1983). 

Although the four quantum numbers n, 1, m, and s, the Pauli Exclusion Principle, and Hund s rules were developed in the context of the Bohr-Sommerfeld model, they all found immediate application to Schrodinger s new quantum mechanics. The first three numbers specified atomic orbitals (replacing Bohr s orbits). Physicist Max Bom (1882-1970) equated the square of the wave functions, to regions of probability for finding electrons in each orbital. Werner Heisenberg (1901-76), whose mathematics provide the foundation of quantum mechanics, developed the uncertainty principle the product of the uncertainty in position (Ax) of a tiny particle such as an atom (or an electron) and the uncertainty in its momentum (Ap) is larger than the quantum (h/47t) ... 

In the paper, the optimization with a risk premium, stochastic optimization and robust optimization were compared. The TMP plant design, even though extremely simplified, had all the characteristics of mixed-integer and dynamical design problems. Therefore, the study offers a suitable application for comparison of the design principles when uncertainty is considered in decision making of optimal design parameters. 

How is the Bohr theory of the hydrogen atom inconsistent with the uncertainty principle (In fact, it was this inconsistency, along with the theory s limited application to non-hydrogen-like systems, that limited Bohr s theory.)... 

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