Elliptical polar coordinates

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

The polarization ellipse along with a designation of the rotation direction (right- or left-handed) fully describes the temporal evolution of the real electric vector at a fixed point in space. This evolution can also be visualized by plotting the curve in (9,tp,t) coordinates described by the tip of the electric vector as a function of time. For example, in the case of an elliptically polarized plane wave with right-handed polarization, the curve is a right-handed helix with an elliptical projection onto the Ocp - plane centered around... 

This is an elliptical equation. At a given position, the endpoint of the electric vector traces out an ellipse on the xy plane, as shown in Figure 2.2. For this reason, the light is said to be elliptically polarized. In the y/ frame where the coordinate axes are along the major axes of the ellipse, the components, Ex and Ey, of the electric vector satisfy the equation... 

The solution of the H2 problem using elliptical polar coordinates and with more elaborate sets of basis functions can be found in ... 

Consider first the quenching of spin-polarized 2s ions in the absence of hyperfine structure or magnetic field effect . Then the 2sj state can decay by allowed Ml transitions, or by electric-field induced El or M2 transitions. An emitted photon is characterized by a propagation vector k and a polarization vector e, and the initial atomic state by a spin-polarization vector P. We choose a coordinate system such that k lies along the z-axis, and write e for the general case of elliptical polarization in the form... 

When 8j and 82 have different phases, the tip of the electric vector traces out an ellipse in any fixed plane perpendicular to k and the wave is said to be elliptically polarized. This general case is roost easily discussed if we choose a coordinate system in which the z-axis coincides with the direction of propagation k and 2 te in the directions Ox, Oy respectively. If the complex amplitudes are... 

Here A is the distance of the foci, which are found on the. s 12-axis. For = 0we have plane polar coordinates. Varying v e [0,2n at constant u describes an elliptical orbit with a = yjA2 + u2 and b = u its semimajor and semiminor axis, respectively. 

Analogous to the spherical filler of radius R in the Kraus model, Bhattacharya and Bhowmick [31] consider an elliptical filler represented by R(1 + e cos 0), in the polar coordinate. The swelling is completely restricted at the surface and the restriction diminishes radially outwards (Fig. 40 where, qt and q, are the tangential and the radial components of the linear expansion coefficient, q0). This restriction is experienced till the hypothetical sphere of influence of the restraining filler is existent. One can designate rapp [> R( 1 + e cos 0)] as a certain distance away from the center of the particle where the restriction is still being felt. As the distance approaches infinity, the swelling assumes normality, as in a gum compound. This distance, rapp, however, is not a fixed or well-defined point in space and in fact is variable and is conceived to extend to the outer surface of the hypothetical sphere of influence. 

K thus defines a static polarization/rotation—whether linear, circular or elliptical—on the Poincare sphere. The 2, r representation of the vector K gives no indication of the future position of K that is, the representation does not address the indicated hatched trajectory of the vector K around the Poincare sphere. But it is precisely this trajectory which defines the particular polarization modulation for a specific wave. Stated differently a particular position of the vector K on the Poincare sphere gives no indication of its next position at a later time, because the vector can depart (be joined) in any direction from that position when only the static 2, r coordinates are given. 

Figure 1 Myoglobin and the heme-CO coordinate system 6 is the angle between C-0 and the heme plane normal and

Figure 4 shows the SCF results in spheroidal coordinates for the excited bending state (0,4) as a function of the coordinate parameter a. For each of the states, the result is compared with the energy given by SCF in polar (hyperspherical) coordinates. Also shown are the results of a bare-mode approximation, a crude model which assumes for the mode a potential = 0), and similarly postulates a separate potential F( cq, > ) for the q mode, without any self-consistency in the treatment of the two modes. It is evident from Fig. 4 that the physically motivated elliptical (spheroidal) SCF modes do better in this case than the hyperspherical coordinates. Also, the SCF correction gives an important improvement on the bare-mode results. Most important, coordinate optimization, that is, imposing condition, (25) yields a noticeably better result than the SCF energy in a spheroidal system that is not refined for the best a value. 

We now turn to the inner-sphere redox reactions in polar solvents in which the coupling of the electron with both the inner and outher solvation shells is to be taken into account. For this purpose a two-frequency oscillator model may the simplest to use, provided the frequency shift resulting from the change of the ion charges is neglected. The "adiabatic electronic surfaces of the solvent before and after the electron transfer are then represented by two similar elliptic paraboloids described by equations (199.11), where x and y denote the coordinates of the solvent vibrations in the outer and inner spheres, respectively. The corresponding vibration frequencies and... 

This fact may be arrived at in two ways. One way is to write down the Schrodinger equation for using spherical polar coordinates or elliptical coordinates. (coordinate systems.) Then one atten ts to separate coordinates and finds that the 4> coordinate is indeed separable from the others and yields the equation... 

A light beam propagating along the x axis of a Cartesian coordinate system is passed through a polarizer oriented in either the y or the z direction. The intensity of the light measured with the y orientation of the polarizer is 1.3 times that measured with z. What is the ellipticity of the light (h) What ellipticity corresponds to complete circular polarization ... 

The lamellar reflections are not flat, but are curved i.e., there is a continuous shift in the z-position of the maxima (z ) in the lamellar peaks as a function of x (Figure 1) Because of this curvature, the two-dimensional (2-D) data could not be fitted in Cartesian coordinates. But they are not curved enough to be a circle, hence the polar coordinates ordinarily used in analyzing the wide-angle x-ray diffraction patterns cannot be used either. It appears that the description in elliptical coordinates provides the best fit to the data. This feature of the scattering curve will be analyzed in detail in this paper. 

Alternative coordinate systems. We have seen how coordinate systems can be cleverly exploited to advantage. For example, consider the elementary log r and 0 solutions for point sources and vortexes obtained in cylindrical polar coordinates. In Chapters 2 and 3, they were rewritten in (x,y) coordinates in order to develop solutions for line fractures and shales. Or consider the conformal mappings introduced in Chapter 5 there, the simple solutions in Chapters 2 and 3 were extended to flows in complicated geometries. A newer, more powerful approach involves the use of boundary-conforming grid sytems that wrap around wells and fractures in the nearfield and at the same time conform to the external boundaries of the farfield. The simplest example is provided by cylindrical coordinates, used to model circular wells concentrically located in circular reservoirs. Another is furnished by elliptical coordinates, used to model flows into straight, finite-length fractures in infinite systems. 

The electronic Schrodinger equation for the hJ ion can be solved by transforming to a coordinate system that is called confocal polar elliptical coordinates. One coordinate is f = (rA + rB)/rAB, the second coordinate isp = (rA — rB)/rAB, and the third coordinate is the angle , the same angle as in spherical polar coordinates. The solutions to the electronic Schrodinger equation are products of three factors ... 

To calculate the flow rate Q for the elliptic channel, we need to evaluate a 2D integral in an elliptically shaped integration region. This is accomplished by coordinate transformation. Let p, 4>) be the polar coordinates of the unit disk, that is, the radial and azimuthal coordinates, which obey 0 < p < 1 and 0 < (p < 2n, respectively. The physical coordinates (y, z) and the velocity field u can then be expressed as functions of p, [Pg.33]

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