Polar representation

Making use of the polar representation of a complex number, the nuclear wave function can be written as a product of a real amplitude, A, and a real phase, S,... 

In the active interpretation, the vector whose cartesian and polar representations are ... 

Wallace, R. (1984), Large Amplitude Vibration in Polyatomic Molecules. I. A Polar Representation of Orthogonal Relative Coordinates, Chem. Phys. 88, 247. 

In problems involving optically active particles it is usually more convenient to use the amplitude scattering matrix in the circular polarization representation. The transformation from linearly to circularly polarized electric field components is... 

Circular dichroism and optical rotation for particulate media may be operationally defined in terms of the Stokes parameters (2.80), which in the circular polarization representation are written... 

Thus, the difference between the diagonal elements of the forward amplitude scattering matrix in the circular polarization representation has a simple physical interpretation. Although we considered identical particles for conveni-... 

Kg- 2. Polar representation of periodic relationships of the elements. (Source Oumibix, U.SAJ) At upper right is shown the conventional representation, see the front matter for the current representation... 

Fig. 2a, b Polar representation of the Jahn-Teller distortion of the C60 molecule obtained with a strong coupling calculation, a Unimodal distortion occurring for n-1, 2, 4, 5. b Bimodal distortion for n=3. (After [4])... 

The spherical-polar representation of an SDF does introduce some difficulties. Its non uniformity results in average structures for different regions of the local space converging at different rates, makes the determination of an appropriate grid size less straightforward, and introduces additional complexity in its visualization. The presence of a pole in the data structure can also be somewhat of an inconvenience. 

In the Appendix we have included examples of code for the accumulation and normalization of a SDF in spherical-polar representation for a molecular system such as water. 

Here, as mentioned above, we have used (22) to combine the two Gaussians in (19) in a single Gaussian function evaluated at zero time. The symbol in this equation indicates that some irrelevant constants have been dropped, not that there are approximations in this result. Below we replace it with an equality as the constants which normalize the propagators divide out when the appropriately normalized correlation function is computed. The semi-classical amplitude can be conveniently re-expressed by introducing a polar representation of the complex polynomials identifying the initial and final occupied mapping states, thus... 

The implementation of the algorithm outlined above is somewhat delicate due to our use of the polar representation of the complex Hermite polynomials that project onto the final states (3 or / . When the complex polynomial is zero, the phase is ill-defined. This is reflected in the expression of the force in (44) by the apparent singularity in the off-diagonal terms. The existence of a divergence in the force, however, depends on the behavior of the gradients of the off-diagonal terms of the electronic Hamiltonian. As they usually are, or go to, zero very rapidly in regions of zero population of the final state, it is... 

Figure 6-24. a) Polar representation (from -90° to +90°) for the angular dependence of fluorescence anisotropy (H-V) in Tg films with grain sizes of 1500 nm (open squares), 600 nm (filled circles), and 200 nm (open circles), b) The angular dependence of electroluminescence anisotropy (H-V) measured from the LED contacted part of the same samples. 

Fig. 26.5. Metallic and magnetic insulating states of a ID chain with one electron and one orbital per site (a) schematic representation of a ID chain, (b) metallic state, (c) magnetic insulating state in non-spin-polarized representation, (d) magnetic insulating state in spin-polarized representation.

SOLUTION In the polar representation, — 1 = e . The three cube roots are... 

In this polar representation, r is called the modulus or magnitude of z (r = z ), and the angle 0 is called the argument or phase of z. Note that as 0 varies from 0 to 2tz, el scribes a circle of unit radius in the complex plane (see Figure A.2), while its real part oscillates as cos 0. 

We are now ready to generalize to spaces of higher dimensionality the well known polar representation of the position vector of a particle in a three-dimensional space. A hyperspherical representation of the Jacobi vectors can be developed, corresponding to the projection on a d-dimensional hypersphere, with d=(n-l)D for n particles in a D-dimensional space after the separation of the center of mass. The d h3rperspherical coordinates consist of a hyperradius, which does not depend on the particular Jacobi set chosen, and d-1 hyperangles, dependent on the arrangement of the particles. 

In three-dimensional space there are two possible symmetric trees (fig.3), the first corresponding to the usual polar representation related to the spherical hrirmonics... 

To sum up, we have derived a boundary condition for a bound (quasi-) stationary trajectory using the proper polar representation r, ict) pr,iE/c)... 

While the FT of the even cosine function is real, the result for sin(27rvot) is imaginary (S(v — vo)- --b Vo))/(2i). Since the sine function is 90° out of phase to the corresponding cosine, it is clear that the imaginary axis is used to keep track of phase shifts, consistent with the polar representation of a complex number x + iy = re with r = Jx - -y and 4> = tan (y/x). In this phasor picture, positive and negative frequencies can be interpreted as clockwise and counterclockwise rotations in the complex plane. 

FIG U RE 9.9 Polar representation of the wave function as a vector with components and Bq, according to Euler s formula. 

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