Square absolute

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

The energy spectral density function (or power spectrum) P f) is given by the absolute square of P f) ... 

Finally, using the physical interpretation of the quantum site-state coefficients ai , we can write down an explicit form for the probability functions p . Since 0 is defined by the list n, we must simply write down the probability that a measurement of the 2r states around a given site F will yield a 2r-tuple which is an element of n. We therefore get sums of products of absolute squares, with individual list elements contributing the terms and list elements... 

Many kinds of transition probabilities depend on DOs. Photoionization cross sections, are proportional to the absolute squares of matrix elements between DOs and continuum orbitals, or... 

The X-ray detector measures the intensity of electromagnetic waves, i.e., the absolute square 2 of their amplitude. Thus, in combination, the upper path between density and intensity through the square is written as... 

The Landau-Zener expression is calculated in a time-dependent semiclassical manner from the diabatic surfaces (those depicted in Fig. 1) exactly because these surfaces, which describe the failure to react, are the appropriate zeroth order description for the long-range electron transfer case. As can be seen, in the very weak coupling limit (small A) the k l factor and hence the electron transfer rate constant become proportional to the absolute square of A ... 

In 4-component KS-DFT spin is no longer a good quantum number because spin-orbit coupling arises. We could circumvent this problem employing an external axis of quantization for the spin, which is conveniently the -axis. Such an approach is called collinear. The -component of the spin operator leads to the spin density and can be calculated by subtracting the a- and /3-spin densities, i.e., the sum of the absolute-squared a-orbitals minus the sum of the absolute-squared / -orbitals, from each other. 

One sees that the only possible values that A can take are the a , each occurring with a probability equal to the absolute square of the projection of j/ on the corresponding eigenvector The case of a continuous spectrum requires some adjustment. 

The fine-structure constant a indicates that first-order perturbation theory has been applied the linear dependence on the photon energy Eph is due to the length form of the dipole operator used in equ. (2.1), and the wavenumber k compensates the 1 /k which appears if the absolute squared value of the continuum wavefunction is used (see equ. (7.29)). The summations over the magnetic quantum numbers M, of the photoion and ms of the photoelectron s spin are necessary because no observation is made with respect to these substates. Due to the closed-shell structure of the initial state with f — 0 and M = 0, the averaging over the magnetic quantum numbers M simply yields unity and is omitted. 

In a next step the absolute squared value of the matrix element has to be evaluated together with the summation over the magnetic quantum numbers M, and ms. This gives... 

Since the energy of the photoelectron is fixed, it is only its direction which counts.) If the absolute squared values of the matrix elements are calculated one gets diagonal terms which can be collected in the expression... 

From the Uncertainty Principle, we no longer speak of the exact position of an electron. Instead, the electron position is defined by a probability density function. If this function is called p (x,y,z), then the electron is most likely found in the region where p has the greatest value. In fact, p dr is the probability of finding the electron in the volume element dr (= dxdydz) surrounding the point (x,y,z). Note that p has the unit of volume-1, and pdr, being a probability, is dimensionless. If we call the electronic wavefunction f, Born asserted that the probability density function p is simply the absolute square of tjr. 

Since ijr can take on imaginary values, we take the absolute square of f to make sure that p is positive. Hence, when jr is imaginary,... 

Finally, Ihk is the absolute square of the all-important structure factor... 

The absolute square in Eq. (3.30.4) implies that the diffraction intensity Ihkii ) does not have an explicit phase and therefore masks the atom positions (x, /, zj),j = 1,2,..., n], the main goal of X-ray structure determination. This "phase problem" frustrated crystallographers for decennia. However, when one compares the experimental data (thousands of different diffraction intensities f a), with the goal (a few hundred atomic position and their thermal ellipsoid parameters B), one sees that this is a mathematically overdetermined problem. Therefore, first guessing the relative phases of some most intense low-order reflections, one can systematically exploit mutual relationships between intensities that share certain Miller indices, to build a list of many more, statistically likely mutual phases. Finally, a likely and chemically reasonable trial structure is obtained, whose correctness is proven by least-squares refinement. This has made large-angle X-ray structure determination easy for maybe 90% of the data sets collected. 

In the Anderson picture the suppression of classical chaotic diffusion is understood as a destructive phase interference phenomenon that limits the spread of the rotor wave function over the available angular momentum space. The localization effect has no classical analogue. It is purely quantum mechanical in origin. The localization of the quantum rotor wave function in the angular momentum space can be demonstrated readily by plotting the absolute squares of the time averaged expansion amplitudes... 

V is the volume of the unit cell and should not be confused with V(r) = V(xyz). From the fact that only the absolute square of F can be measured, the phase of the complex value of F is lost. This is the phase problem of crystallography, which found its solution by the introduction of the method of isomorphous replacement in the fifties... 

The differential cross section is given by the absolute square of this amplitude (4.48). Trial values of the phase shifts Sl are then varied to give... 

To understand an electron—atom collision means to be able to calculate correctly the T-matrix elements for excitations from a completely-specified entrance channel to a completely-specified exit channel. Quantities that can be observed experimentally depend on bilinear combinations of T-matrix elements. For example the differential cross section (6.55) is given by the absolute squares of T-matrix elements summed and averaged over magnetic quantum numbers that are not observed in the final and initial states respectively. This chapter is concerned with differential and total cross sections and with quantities related to selected magnetic substates of the atom. 

The target part of the entrance-channel state Ovoko) is a coherent superposition of magnetic substates defined by an arbitrary choice of coordinate frame. It is transferred into another fully-coherent superposition of states in the exit channel ivjkj). The scattering amplitude is defined as a generalisation of (4.46), so that its absolute square is the corresponding... 

The divergent phase vanishes when the absolute square of (10.37) is taken for the diflFerential cross section (6.60), which has the form... 

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