Legendre operator

A wavefunction Yi m for a specific state of orbital angular momentum, i.e., orthogonal functions of the angular coordinates which satisfy the differential equation = —1(1 + 1)K, where is the Legendre operator. The functions are polynomials in sin 6 and cos. Spherical harmonics are the angular factors in centrosymmetric atomic orbitals. 

In order to perform this operation, we expand the wavepacket in terms of normalized associated Legendre polynomials... 

The state variables are (41). The time evolution (63) does not involve any nondissipative part and consequently the operator L, in which the Hamiltonian kinematics of (41) is expressed, is absent (i.e., L = 0). Time evolution will be discussed in Section 3.1.3. We now continue to specify the dissipation potential 5. Following the classical nonequilibrium thermodynamics, we introduce first the so-called thermodynamic forces (X 1-.. X k) Jdriving the chemically reacting system to the chemical equilibrium. As argued in nonequilibrium thermodynamics, they are linear functions of (nj,..., nk,) (we recall that n = (p i = 1,2,..., k on the Gibbs-Legendre manifold) with the coefficients... 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

From a series of transformations of Equation 1 we obtain a new partition function (T) whose independent variables are temperature, pressure, solvent mole number, and the chemical potentials of the solutes (components 2 and 3). These transformations consist of first creating a partition function with pressure rather than volume as an independent variable, and then using this result to create yet another partition function in which we have switched independent variables from solute mole numbers to solute chemical potentials. These operations are analogous to the Legendre transforms commonly employed in thermodynamics. 

T vanishes for T O, because the Z axis is an axis of symmetry. o = Pz, (cos6) is the Legendre pol5momial or axial spherical harmonic and r is the absolute value of radius vector of the electron. The term L=0 does not appear in Eq. (27) because for this term the barycenter rule does not apply, as it ought to, since we are concerned with the barred operator [cf. Ref. (15), legend to Table 16]. 

The information on the P" given in 5.5 will be used again here however only (the Legendre polynomials ) need to be used. In addition to the information given earlier, we will need to have the result of the fle-operator operating on the... 

Simplifications. In the form we give to tp, the use of half integers, which is a complication, is avoided. Only the integers m G Z that appear in the associated Legendre polynomials and P 1 1 are employed. Half integers m = m + 1/2 appear, for example, in the formula implying the total angular momentum operator of the electron (see Appendix C) and will be introduced in the Zeeman-effect (Chap. 14). 

Before 1970 the multipole expansion (by which we mean the expansion in powers of MR) of the interaction operator was usually truncated after the R dipole-dipole term, so that the only dispersion interaction term was —C R. Around 1970 it became clear that this approximation was not sufficient and that more terms were needed. However, the straightforward application of the Taylor expansion, and its natural formulation in terms of Cartesian tensors [77], soon becomes cumbersome. Nineteenth century potential theory [78,79] came to the rescue. In this theory the multipole series is rephrased in terms of associated Legendre functions, which enables a closed form of it. Multipole operators are defined as... 

Here y m are the spherical harmonic functions Q m = yj47r/(2k + 1) y, m is the Racah tensor operator = rk Ykm is the irreducible tensor operator Pk m (not to be confused with the Legendre polynomials) are unnormalised homogeneous polynomials of Cartesian coordinates proportional to the function rk Ykm + Yk m) Ok are referred to as equivalent operators which are constructed of only the angular momentum operators. 

The angular dependence of the stream function represents one of the Legendre polynomials that is unaffected by the operator for creeping viscous flow in spherical coordinates. In other words,... 

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