Square root of an operator

In the remaining sections of this chapter you will be multiplying and dividing exponentials, taking the square root of an exponential, and working with logarithms. We will furnish brief comments on these operations as we come to them. For more detailed instructions, see Appendix I, Parts B and C. 

Equation (13) is the first important equation for open tubular column design. It is seen that the optimum radius, with which the column will operate at the optimum velocity for the given inlet pressure, increases rapidly as an inverse function of the separation ratio (cc-1) and inversely as the square root of the inlet pressure. Again it must be remembered that, when calculating (ropt)5 the dimensions of the applied pressure (P) must be appropriate for the dimensions in which the viscosity (r)) is measured. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

The expectation value of H in the coherent state (7.17) can be evaluated explicitly for any Hamiltonian. However, an even simpler construction of Hd (valid to leading order in N) can be done (Cooper and Levine, 1989) by introducing intensive boson operators (Gilmore, 1981). In view of its simplicity, we report here this construction. If one divides the individual creation and annihilation operators by the square root of the total number of bosons, the relevant commutation relations become... 

The mathematical notion of an operator may be unfamiliar it is a rule for modifying a function. A comparison of the ideas of operator and function may be useful Whereas a function acts to take an argument, called the independent variable, as input, and produces a value, called the dependent variable an operator takes 2l function as input and produces a function as output. Multiphcation of a function by a constant, taking a square or square root, differentiation or integration, are examples of operators. Table 8.1 contains examples of functions and operators. 

An estimate of compound random errors is obtained from the square root of the sum of the squares of the RSDs attributed to each component or operation in the analysis. If the analysis of paracetamol described in Box 1.3 is considered then, assuming the items of glassware are used correctly. Assuming the items of glassware are used correctly the errors involved in the dilution steps can be simply estimated from the tolerances given for the pipette and volumetric flasks. The British Standards Institution (BS) tolerances for the grade A glassware used in the assay are as follows ... 

A more conventional mechanism appears to be operative in the photopolymerization of ethyl acrylate [178] and methyl methacrylate [179] in aqueous solution, sensitized by fluorescein and Erythrosin, respectively. Ascorbic acid is the reducing agent in both cases and it is observed that the reaction does not proceed in the absence of buffer, usually phosphate buffer pH 6. Polymer formation starts after an induction period but its dependence on light intensity and ascorbic acid concentration has not been determined. The rate of photopolymerization is proportional to the monomer concentration and to the square root of the light intensity, dye, and ascorbic acid concentration. The authors report the order with respect to the monomer as 3/2. However, from our analysis of the data for fluorescein, which are more... 

In the IR spectral region the noise characteristics, typical of the measurement system, have led to the development of multiplex detectors whose operation is based on the use of mathematical transformations such as the Fourier and the Hadamad. With these detection systems, an appropriate encoding (transformation) of the raw spectral signal is performed, which permits simultaneous monitoring of a wide spectral window with a single-channel sensor. A SNR (multiplex) advantage is achieved that is typically proportional to the square root of either the observation time or the number of spectral-resolution elements contained in that spectral window. 

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