Statistical case

Figure A3.4.2. A simple illustration of limiting dynamical behaviour case 1 statistical, case 2 coherent (after [20]).

To provide a definition of the density matrix in terms of fundamental wave-functions first consider the generalization of the expectation value from quantum mechanics to quantum statistical mechanics. In the quantum statistical case, an additional average over the probability density needs to be considered in the calculation of the expectation value ... 

Peck, R., Haugh, L., and Goodman, A., Statistical Case Studies A Collaboration Between Academe and Industry, Student Edition... 

Czitrom, V. and Spagon, P. D., Statistical Case Studies for Industrial Process Improvement... 

After reading Lipsitt s descriptive details, examples, and analysis, do you feel you now have a general understanding of a new term If the writer were to expand his definition, what might he add to make your understanding even more complete More statistics Case studies Testimony from doctors or patients themselves ... 

To conclude this discussion on IDLs, it is useful to compare Eqs. (2.19) and (2.29). Both equations relate Xp to a ratio of a standard deviation to the slope of the least squares regression line multiplied by a factor. In the simplified case, Eq. (2.19), the standard deviation refers to detector noise found in the baseline of a blank reference standard, whereas the standard deviation in the statistical case, Eq. (2.29), refers to the uncertainty in the least squares regression itself. This is an important conceptual difference in estimating IDLs. This difference should be understood and incorporated into all future methods. 

It is important to note that T is given by the same formula as in the statistical limit, but its physical meaning is different. F corresponds to a half-width of a band of discrete levels, while y " is a width of a single level. Nevertheless, there is a continuous transition between the strongcoupling and the statistical cases if the /-level density or /-level width is increased, the / manifold tends to a continuum, so that a single, broad resonance remains with the width F =... 

The short-time behavior of the system is essentially the same as in the statistical limit, but differs by the residual s character of the finally attained mixed states. In the statistical case, however, the s character of the initially excited state is irreversibly lost. This difference is due to a discrete-level structure of the /i manifold in the strong-coupling case, as compared to the statistical quasi-continuum. The deviation from the purely statistical behavior will therefore be attenuated, when /-level density or /-level width increases. In the first case, since N p, the relative intensity of the long component in decay ( 1/N) will be reduced. In the latter, not only its amplitude but also its decay time decreases because of the weak-mixing effect. In both situations, the detection of the long decay component becomes more and more difficult when a strong /-level overlapping transforms the discrete / manifold into a statistical quasi-continuum. 

The case ascertainment depends largely on the sources of data that are used, such as morbidity statistics or observational studies. Mortality statistics are unhelpful because contact dermatitis commonly is never fatal. In morbidity statistics, case ascertainment usually involves registration of persons with dermatitis who fulfil additional criteria for registration, such as hospital admission, sickness leave, or referring to a specialist. This restriction in the definition of a case will probably result in selective inclusion of the more severe cases, since a large proportion of individuals suffering from contact dermatitis do not come to medical attention or are seen elsewhere (Fig. i). 

The data are arranged in a matrix form the column vectors are called variables and the row vectors are called mathematical-statistical cases (objects or samples) ... 

Grouping variable (Y) shows whether a given object (statistical case) belongs to a certain class (e.g., the code 0 means ill and the code 1 means healthy) it is also called dummy variable. 

A position measurement provides an important example of the statistical case. Consider a particle that moves parallel to the x axis. Assume that we make a set of position measurements with the state of the system corresponding to the same wave function, 4r(x, t), at the time of each measurement. The outcome of any measurement must be an eigenvalue of the position operator. The eigenfunction of the operators is the Dirac delta function in Eq. (16.3-36). Any value of x can be an eigenvalue, denoted by a in Eq. (16.3-36). Any values of x can be an outcome of the position measurement. 

Distinguishing the Predictable Case from the Statistical Case... 

In the predictable case all outcomes of repeated measurements will be equal to each other and to the mean value. The standard deviation will equal zero. In the statistical case the outcomes will vary and the standard deviation will be nonzero. 

The fact that the standard deviation is nonzero shows that the statistical case applies. Inspection of the probability distribution also reveals this fact, since it is nonzero at more than one point. 

This example shows that the statistical case applies if the wave function just prior to the measurement is not an eigenfunction of the operator corresponding to the variable being measured. 

If the statistical case applies we want to be able to predict the uncertainty in a proposed measurement. If a single measurement is to be made, there is roughly a two-thirds probability that the result will lie within one standard deviation of the expectation value. We will use the standard deviation as a prediction of the uncertainty. In Example 16.16 we determined that the standard deviation of the position of a particle in a one-dimensional box of length a is equal to 0.180756a for the n = 1 state. We now find the standard deviation of the momentum for this state. 

There is roughly a two-thirds probability that the momentum lies between —h/la and h/la. The statistical case applies, as it did with the position. We can predict the mean and the standard deviation of a set of many measurements of the momentum, but it is not possible to predict the outcome of a single measurement. 

The uncertainty principle is a rather subtle concept, and deserves more discussion than we give it in this book. However, the main idea is that it requires that the statistical case applies to at least one of a conjugate pair of variables, and if the predictable case applies to one of the variables, the other variable has an infinite uncertainty. 

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