Legendre functions, spherical harmonics

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

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,... 

Since spherical harmonics are functions from the sphere to the complex numbers, it is not immediately obvious how to visualize them. One method is to draw the domain, marking the sphere with information about the value of the function at various points. See Figure 1.8. Another way to visualize spherical harmonics is to draw polar graphs of the Legendre functions. See Figure 1.9. Note that for any , m we have F ,m = , the Legendre function carries all the information about the magnitude of the spherical harmonic. 

The 4>) function turns out to be an exponential and the ( ) function consists of Legendre polynomials. Their product () ( ) gives the spherical harmonic functions which Arfken writes as Y 6, ). Then, from Eq. 20.56,... 

The three-particle distribution function g3(r,s) can be expressed in a series of Legendre polynomials [63]. Then expressing the Legendre polynomials in terms of spherical harmonics, we can write the expression for g3(r,s) as... 

The functions Qim(9) and consequently the spherical harmonics Yim(6, associated Legendre polynomials, whose definition and properties are presented in Appendix E. To show this relationship, we make the substitution of equation (5.42) for cos 6 in equation (5.51) and obtain... 

The sum over wave numbers is written as an integral over wave number spaee, with a density of states in wave number space of 2/(2rc)- = Naii/ 6n ). The exponent k d is simply kd cos 0, and the spherical harmonics can be written out in terms of the associated Legendre functions, PtJ (cos (Schiff, 1968, p. 80),... 

The functions lVm(r) are the spherical harmonics, which are given in terms of the associated Legendre polynomials P m(x). For m > 0... 

All the correlation functions above are normalized, therefore equations (4 and 5) are identical to correlation functions over linear momentum p = mv and angular momentum J — lu, respectively. Note that, in this context I is the moment of inertia tensor The correlation function in equation (6) is calculated over the spherical harmonics. If m = 0, this reduces to time correlation function over Legendre polynomials ... 

A linear combination of the bound and continuum atomic wavefiinction was used to approximate the continua (10). Wavefunctions in the spherical potential were separated into the spherical harmonics and the radial wavefunctions. The spherical harmonics are expressed in terms of the associated Legendre functions. The differential equation for the radial wavefunction R at position r is... 

In such cases, Y is a spherical harmonic and w is a product of a power of r, an exponential function, aud a Legendre polynomial in r. We want to take a snapshot so time is fixed and we only care about the spatial coordinates. If we want to visualize these solutions, it is useful to think about where the functions are zero and what sorts of symmetry they have. 

On the other hand, an expansion of the Legendre polynomials over spherical harmonic functions is known in the following form... 

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. 

Other advantages of working in terms of spherical harmonic functions are that for cases with fibre symmetry, the Legendre addition theorem can be used, and affords considerable algebraic simplifications (see for example Ref. 25), and that for lower symmetries, the treatment can readily be generalised. It should be mentioned that the exact definitions of P2 cos9), etc., and p 9), can differ in different treatments due to the adoption of different normalisation procedures (see, for example. Chapter 5, Section 5.2).)... 

Assuming a spherical FS the appropriate basis function for the SC order parameter are spherical harmonics of angular momentum I. The interaction in the l-v/svc chaimel is given by (Pi = Legendre polynomial)... 

In some early work, an expansion in powers of sin d has been employed. However, the spherical harmonics form an orthogonal basis and are thus more appropriate. An expansion in Legendre polynomials P (cos 0) has also been frequently used. Although these functions are equivalent to the Y%0,4>)> due account of the scaling factor has not always been included in previous comparisons of anisotropy results. The microscopic anisotropy parameters comprise the terms of various physical origins which enter into the hamiltonian for the system. If the hamiltonian is written in a representation Slf(0,4>) in which the quantization axis is along the magnetization direction, the microscopic and... 

To solve the eigenvalue problem, the wavefunction is expanded in a series of spherical harmonics Yj ( < >) But now the hamiltonian commutes with Thus mj is a good quantum number. If we assume that in the ground state the complex has a linear equilibrium geometry (like Hg-HCl and Mg-HF (i 7,18)), Then only the state corresponding to mj = 0 will be populated, leading to the excitation of states with mj = 0 only. We thus choose Yjo (0,0) as basis functions, and write V(0) as a linear combination of Legendre polynomials ... 

Legendre polynomials (7.21) may be used for this purpose, and the procedure involved is identical to that used in Sec. 7.2c for the series representation of the function t)(Mo l ) carrying out this calculation it is convenient to introduce also the spherical harmonics F (Q) defined by... 

These wavefunctions are functions that were well known to the people who developed quantum mechanics. They are called spherical harmonics and are labeled Yi, (or Once again, classical mathematics anticipated quantum mechanics in the solution of differential equations. Although the Legendre polynomials do not distinguish between positive and negative values of the quantum number m, the exponential part of the complete wavefunction does. Each set of quantum numbers ( , m ) therefore indicates a unique wavefunction, denoted that can describe the possible state of a particle confined to the surface of a sphere. The wavefunction itself does not depend on either the mass of the particle or the radius of the sphere that defines the system. 

The function (0, spherical harmonics where P (0) are the normalized Legendre polynomials ... 

The properties of spherical harmonics and the associated Legendre functions are given in ... 

The expansion in Legendre polynomials or more generally spherical harmonics is chosen because they are orthogonal functions. The coefficients Ul in the expansion can be obtained by multiplying both sides of Eq. (24) by Pl(cos0) and integrating over 0, with the result ... 

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. 

There are an infinite number of solutions to these equations, and two integers are needed to label and distinguish the different spherical harmonic functions. The conventional choices for these two integers are I and m. The 0 dependence of these functions is expressed with a special set of polynomials called the associated Legendre polynomials. 

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