Spheroidal quantum numbers

Figure 1. Degenerate energy eigenstates for each n sorted according to m = 0,1,2,. ..(<7, 7t,5, and according to eigenvalues of the separation constant A (designated by the spheroidal quantum numbers n and n ), thereby specifying spheroidal eigenstates as hybrids of the usual spherical eigenstates.

The gamma functions Ak(p) and Bj(pt) may be obtained by the use of recursion formulas an extensive tabulation is due to Flodmark (141). In the case of Slater orbitals of principal quantum number 4 or 6, application of Slater s rules leads to nonintegral powers of r in the radial wave function consequently, changing to spheroidal coordinates introduces A and B functions of nonintegral k values, that is, incomplete gamma functions. These functions can, however, be computed (56, 57) and the overlap... 

Symmetrical tops are of two types. A prolate spheroid (football shape) in which /A < /b, and an oblate spheroid (pancake shape) in which /A > /b. Again, there are 2J + 1 sets of energy levels for each J, but they are no longer degenerate, as can be seen from equation (A6.9), which includes both the J and K quantum number. 

As far as the theoretical description is concerned, the confined acoustic phonons in an elastic sphere were first theoretically studied by Lamb [29]. He derives two types of confined acoustic modes, spheroidal and torsional modes. The frequencies of these two modes are proportional to the sound velocities in particles and inversely proportional to the particle size. The spheroidal mode is characterized by the quantum number / > 0, while the torsional modes are characterized by / > 1. From the symmetry arguments, Raman-active modes are spheroidal modes with I = 0 and 2. The I = 0 mode is purely radial with spherical symmetry and produces totally polarized spectra, while the Z = 2 mode is quadrapolar and produces partially... 

The total effective potential determining the electronic motion via the Kohn-Sham equations is expected to be spheroidal as well. Therefore all spherical shells n, /, m are expected to split into spheroidal subshells m, p,k. Here m is the preserved azimuthal quantum number. For time-reversal symmetry only its magnitude m counts p is the parity and k just enumerates the levels of a certain symmetry. The reduced spheroidal symmetry lifts the spherical degeneracy as depicted in Figure 1.11 for Na in the size-range of N from 3 to 18. 

With the knowledge of the elastic constants of the matrix and the dot shape, it is possible to determine the dot radius from the wave number of the spheroidal acoustical mode [52,167,168] The quantum dots are considered as elastic bodies, which are vibrating with eigenmodes, whose frequencies are proportional to their diameter. The eigenfrequencies of the torsional and the spheroidal modes are then given by [162-164]... 

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