Spherical harmonic functions, momentum

Associated with each operator realization of a Lie algebra we generally have a vector space on which these operators act. For the realization given by L this might be either an abstract space of angular momentum states lm), 1 = 0, 1,... m = —/, — l + 1,..., l or a concrete realization of them as spherical harmonic functions Ylm(6, (j)). We can then consider the matrix elements of the operators with respect to this vector space of states and this leads to the important concept of a matrix representation of a Lie algebra. 

In the Arthurs-Dalgarno (31) space-fixed representation, a= j,i and (R,r) is the total angular momentum function, defined as an a pTop riate linear combination of products of spherical harmonic functions for the rotation of R and r (10). Alter-... 

The nuclear quadrupole moment is an expectation value with respect to the nuclear wave function. For the nuclear ground state the nuclear wave function depends upon a radial parameter a, the nuclear spin quantum number / and its projection Mj so that the corresponding ket-vector is denoted as a, I, Mj). The properties of the nuclear spin (in general, an angular momentum) are well known and they can be fully exploited in expressing such an expectation value. For this purpose let us rewrite the electrostatic interaction energy, making use of the expansion in terms of the spherical harmonic functions... 

The angular Schrodinger equation describes the quantisation of the angular momentum and its projection its solutions are represented by the set of spherical harmonic functions Yt m(, other atoms. The radial equation has solutions in the form of Laguerre polynomials... 

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. 

Fig. 1.5 Spherical harmonic functions, F , where / and mi are eingular momentum emd magnetic quantum numbers...

The spherical harmonic functions Y(0,angular momentum (1 ) for a particle in a central field, according to the equation... 

The left-hand side is the operator for the Lj component of the angular momentum acting on the spherical harmonic function (see Further Reading section of this appendix). We can apply this operator to the general form of the spherical harmonic solutions ... 

The angular functions K/m(0, 4>) are called spherical harmonic functions. These functions are eigenfunctions of the operator for the square of the orbital angular momentum and its z component, with eigenvalues given by... 

The Y factor represents the same spherical harmonic functions as in Chapter 17, which are eigenfunctions of Z , the operator for the square of the angular momentum. Substitution of tArei = R r)Y 9,) into the time-independent Schrodinger equation and division by R r)Y 0,(p) gives Eq. (17.2-5) ... 

By convention, we choose the spherical harmonic functions such that they are eigenfunctions of the z-component operator and not the x- or y-component operator. There is nothing special about the z-direction in space. Rather, since we know that there is one direction in which the angular momentum component is quantized (i.e., the wavefunctions are eigenfunctions of one component operator), the convention is that that direction, whatever it happens to be, becomes the z-direction. 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

In the partial wave theory free electrons are treated as waves. An electron with momentum k has a wavefunction y(k,r), which is expressed as a linear combination of partial waves, each of which is separable into an angular function Yi (0. ) (a spherical harmonic) and a radial function / L(k,r),... 

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

It is not possible to ascribe a definite value of the orbital angular momentum to a photon state since the vector spherical harmonic YjM may be a function of different values of . This provides the evidence that, strictly speaking, it... 

The problem is not simplified by Eq. (15), since there exists a closed-form expression for the multi-scattering matrix for n spheres in terms of spherical Bessel and Hankel functions, spherical harmonics and 3j-symbols, where l, l and to, m are total angular momentum and z-projection quantum numbers, respectively (Henseler, Wirzba and Guhr, 1997) ... 

In this section, we will examine the role of interelectronic repulsion in the perspective of the internal symmetries of the shell. The key observation is that in a d-only approximation — i.e. if the t2g-orbital functions can be written as products of a common radial part and a spherical harmonic angular function of rank two - the interelectronic repulsion operator and the pseudo-angular momentum operators commute [2]. This implies that the dominant part of the... 

It is convenient to use spherical polar coordinates (r, 0, ) for any spherically symmetric potential function v(r). The surface spherical harmonics V,1" satisfy Sturm-Liouville equations in the angular coordinates and are eigenfunctions of the orbital angular momentum operator such that... 

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