Angular momentum Legendre functions

In order to transform to the body-fixed representation, we will need to relate the angular functions Wj (R,r) to angular functions defined relative to the body-fixed axes [L., J,K,M,p)QjK ), where J,K,M,p) are the parity-adapted total angular momentum eigenfunctions of Eq. (4.5) and x(0) normalized associated Legendre polynomials of the body-fixed Jacobi angle]. 

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 ... 

The resulting wavefunctions, R(r), 0(8), and associated Laguerre polynomial, an associated Legendre polynomial, and a z-component of the angular momentum, respectively. These functions will not be described in more detail in this section. In the case of the hydrogen atom it is possible to calculate exactly the wavefunctions and the energy eigenvalues. However, it becomes extremely difficult to calculate precisely the... 

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. 

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)... 

The crystal field potential W cf is developed in Legendre polynomials, and with the help of the Wigner-Eckhart theorem (Edmonds, 1957), can be expressed in terms of operator equivalents O which are functions of angular momentum operators J J+, J- This method of Stevens (1952) is described in detail by Hutchings (1966), whose nomenclature is adopted here. The hamiltonian as a function of angular momentum operators is... 

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. 

Relationships among the associated Legendre functions contained in the terms [see Eq. (3.3.15)] cause the integral in Eq. (3.3.21) to be nonzero only if J = /" 1. This selection rule, A/ = 1, means that the angular momentum... 

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