Spherical polynomials

What have we found thus far A magical integer, n, which tells about the space of spherical polynomials of degree n. Another magical integer, k = j + n + 1 tells us the eigenvalue... 

Proportionality factors between spherical operators and uniform spherical polynomials... 

Anderson, J. M., Spherical polynomials. Introduction to Quantum Chemistry, W. A. Benjamin, Inc., New York, 1969, Appendix 2, p. 322. Also private communication with Prof. Anderson. 

The interaction energy can be written as an expansion employing Wigner rotation matrices and spherical hamionics of the angles [28, 130], As a simple example, the interaction between an atom and a diatomic molecule can be expanded hr Legendre polynomials as... 

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

The spherical harmonics are defined in terms of the associated Legendre polynomials, of variable cos 6, and exponential functions in... 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

Relationship of spherical harmonics to associated Legendre polynomials... 

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

To motivate our next step recall that the LOFF spectrum can be view as a dipole [oc Pi (a )] perturbation of the spherically symmetrical BCS spectrum, where Pi(x) are the Legendre polynomials, and x is the cosine of the angle between the particle momentum and the total momentum of the Cooper pair. The l = 1 term in the expansion about the spherically symmetric form of Fermi surface corresponds to a translation of the whole system, therefore it preserves the spherical shapes of the Fermi surfaces. We now relax the assumption that the Fermi surfaces are spherical and describe their deformations by expanding the spectrum in spherical harmonics [17, 18]... 

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

According to the results, it is determined that the asphericities can be described in terms of polynomials in Forni et al. [140] also used an off-lattice model and an MC Pivot algorithm to determine the star asphericity for ideal, theta, and EV 12-arm star chains. They also found that the EV stars chains are more spherical than the ideal and theta star chains. In these simulations the theta chains exhibit a remarkable variation of shape with arm length, so that short chains (where core effects are dominant for all chains with intramolecular interactions) have asphericities closer to those to those found with EV, while longer chains asymptotically approach the ideal chain value(see Fig. 10). 

Here, the permutations of j, k,l,... include all combinations which produce different terms. The multivariate Hermite polynomials are listed in Table 2.1 for orders < 6. Like the spherical harmonics, the Hermite polynomials form an orthogonal set of functions (Kendal and Stuart 1958, p. 156). 

Since the associated Legendre polynomials (and the spherical harmonics) form an orthogonal set, only terms with / = / and m = m do not vanish in the integral of Eq. (3.41). Furthermore, for the 6 integration,... 

For clarity, in this chapter, we discuss the m=0 states in detail. The m 0 states are discussed in an example, although the formalism is equally applicable to all /, m. For m = 0 states, the spherical harmonics Yi reduce to Legendre polynomials. Pi ( cos 6). The first three of them are ... 

Physicists are familiar with many special functions that arise over and over again in solutions to various problems. The analysis of problems with spherical symmetry in P often appeal to the spherical harmonic functions, often called simply spherical harmonics. Spherical harmonics are the restrictions of homogeneous harmonic polynomials of three variables to the sphere S. In this section we will give a typical physics-style introduction to spherical harmonics. Here we state, but do not prove, their relationship to homogeneous harmonic polynomials a formal statement and proof are given Proposition A. 2 of Appendix A. 

Relating the spherical harmonic functions introduced here to the homogeneous harmonic polynomials is not logically necessary in this book. Morally, however, the calculation is well worth doing, in the name of better communication between mathematics and physics. Because this calculation is a bit tricky, we have postponed it to Appendix A. 

In Section 7.1 we will use this characterization of homogeneous harmonic polynomials as a kernel of a linear transformation (along with the Fundamental Theorem of Linear Algebra, Proposition 2.5) to calculate the dimensions of the spaces of the spherical harmonics. 

Note that because the action of SU 2) on is linear and invertible, the transformation R (g) preserves polynomial degree. These representations are related to the spin of elementary particles as we will see in Section 10.4 in particular, P corresponds to a particle of spin-n/2. Spin is a quality of particles that physicists introduced into their equations to model certain mysterious experimental results we will see in Chapter 10 that spin arises naturally from the spherical symmetry of space.)... 

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