Directional quantization

Richtungsanderung,/. change of direction, richtungslos, a. directionless. Richtungsquantelung, /. directional quantization. 

A new quantum number, I, appears the energy is quantized, and there is a new relationship between the quantum number for total angular momentum I, and the quantum number for rotation in the x-y plane, m. For any given value of the total angular momentum, its projection on the z-axis can only have a number of discrete values, so equation 3.21b is a vectorial, or directional, quantization. For example, for I = 1, the total momentum is h /2, and the z-component can only be -h, 0, or +h. Calling a the angle between the angular momentum and its z-component, the only allowed values are cos(a)= 0, or + l 2. 

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

Translational energy, which may be directly calculated from the classical kinetic theory of gases since the spacings of these quantized energy levels are so small as to be negligible. The Maxwell-Boltzmann disuibution for die kinetic energies of molecules in a gas, which is based on die assumption diat die velocity specuum is continuous is, in differential form. 

Raum-mass, n. measure of capacity or volume, cubic measure, -menge,/. amoimt of space, volume, -meter, m. cubic meter, -modell, n. space model, -orientierung,/. orientation in space, -quantelung, /. spatial quantization. -richtung,/. direction in space, -strahl,... 

The Tetrahedral Carbon Atom.—We have thus derived the result that an atom in which only s and p eigenfunctions contribute to bond formation and in which the quantization in polar coordinates is broken can form one, two, three, or four equivalent bonds, which are directed toward the corners of a regular tetrahedron (Fig. 4). This calculation provides the quantum mechanical justification of the chemist s tetrahedral carbon atom, present in diamond and all aliphatic carbon compounds, and for the tetrahedral quadrivalent nitrogen atom, the tetrahedral phosphorus atom, as in phosphonium compounds, the tetrahedral boron atom in B2H6 (involving single-electron bonds), and many other such atoms. 

Thus we have shown that when s and p orbitals are available and s—p quantization is broken an atom can form four (or fewer) equivalent bonds which are directed towards tetrahedron corners. To the approximation involved in these calculations the strength of a bond is independent of the nature of other bonds. This result gives us at once the justification for the tetrahedral carbon atom and other tetrahedral atoms, such as silicon, germanium, and tin in the diamond-type crystals of the elements and, in general, all atoms in tetrahedral structures. 

Some years later a more thorough discussion of the motion of pairs of electrons in a metal was given by Cooper,7 as well as by Abrikosov8 and Gor kov,9 who emphasized that the effective charge in superconductivity is 2e, rather than e. The quantization of flux in units hc/2e in superconducting metals has been verified by direct experimental measurement of the magnetic moments induced in thin films.10 Cooper s discussion of the motion of electron pairs in interaction with phonons led to the development of the Bardeen-Cooper-Schrieffer (BCS) theory, which has introduced great clarification in the field of superconductivity.2... 

As an illustration, the base vector x in Fig. 4 is fixed as x = (1,0) and u = D x rotates in response to changes in A0 and T around the cycle. Thus on completion of the cycle, u/ = D/ x, where the subscript i has been added to indicate the direction of x in the initial lattice. Similarly, the projection of the final vector x/ onto the original lattice is given by X/ = D,Uy, where D, is the tangent matrix before translation. The overall change in Bohr quantized lattice vectors is therefore given by x/ = Mx where the monodromy matrix is given by... 

Among atomic orbitals, s orbitals are spherical and have no directionality. Other orbitals are nonspherical, so, in addition to having shape, every orbital points in some direction. Like energy and orbital shape, orbital direction is quantized. Unlike footballs, p, d, and f orbitals have restricted numbers of possible orientations. The magnetic quantum number (fflj) indexes these restrictions. 

As mentioned in Section Wl, an electron has magnetism associated with a property called spin. Magnetism is directional, so the spin of an electron is directional, too. Like orbital orientation, spin orientation is quantized Electron spin has only two possible orientations, up or down. The spin orientation quantum number )... 

HPLC is often reported to be the technique of best choice for the quantification of food colorants. According to European Directive 94/36/EC, the quantities of synthetic colorants to be added to foods are restricted and thus reliable methods for their quantification must be established. Approved colorants, defined by E-coded numbers (Table 6.6.2), are permitted for non-alcoholic beverages, confectionery products, and even for caviar (dying fish roe). For example, a specific HPLC chromatographic method for the quantization of 14 synthetic food colorants belonging to azo dye, triphenyhnethane, or quinophthalone classes (E 102,104, 110, 122,123, 124, 127, 128, 129, 131, 132, 133, 142, 151) was reported to check their contents in caviar. ... 

In an experiment of thoughts the 3D piece of metal shall be reduced in the z-direction to only a few nanometres, comparable with the electronic de Broglie wavelength (t4 = k), whereas in x- and y-direction it is kept infinite a 2D quantum well is formed. Compared with the former 3D situation, the electrons in x-and y-direction can still freely move in these directions, but not in the z-direction. Electrons in this direction are confined like in a box. The states are quantized, whereas in x- and y-direction the situation does not differ from that in the 3D case. Figure 5a shows the quantized situation in z-direction with well-defined Ak values n— I, 2, etc., and Figure 5b indicates the Ak values close to 0. 

The Ak values in y- and z-direction are quantized, but not in x-direction, resulting in a parabola with Ak O. The density-of-states within the quantum wire results in hyperbolas for each individual ky- and fc -state. 

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