Angular momentum spherical harmonics

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... 

The orbitals d. and dy can be expressed in terms of the complex forms di and whose angular parts are given by the spherical harmonics and respectively. The matrix of orbital angular momentum about the z axis in the complex basis is... 

Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which A/ = 1 and for which Am = 0, 1. Thus, an observed spectral line vq in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers vq + (/ bB/he), vq, and vq — (HbB/he). 

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. 

The use of spherical harmonics in real form is limited by the fact that, for m 0 they are not eigenfunctions of Lz. They may be used to specify the angular distribution of electron density, but at the expense of any knowledge about angular momentum, and vice versa. 

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... 

If j = 0 there is only one vector spherical harmonic which is identical with the longitudinal harmonic Y 1 = nVoo- From this observation it follows that there are no transverse spherical harmonics for j = 0. It also means that the state with angular momentum zero represents a spherically symmetrical state, but a spherically symmetrical vector field can only be longitudinal. Thus, a photon cannot exist in a state of angular momemtum zero. 

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 the following, we pay special attention to the connections among the spherical, Stark and Zeeman basis. Since in momentum space the orbitals are simply related to hyperspherical harmonics, these connections are given by orthogonal matrix elements similar (when not identical) to the elements of angular momentum algebra. 

If the states have different angular momentum character then the angular integration over the spherical harmonics guarantees orthogonality. But if the states have the same angular momentum character then the orthogonality... 

Spherical harmonics are the eigenfunctions of the orbital-angular-momentum operators /. We shall give them the symbol Yim... 

Some readers may wonder why we make this restriction, especially if they have experience applying angnlar momentnm operators to discontinuous physical quantities. It is possible, with some effort, to make mathematical sense of the angular momentum of a discontinuous quantity hut, as the purposes of the text do not require the result, we choose not to make the effort. Compare spherical harmonics, which are effective because physicists know how to extrapolate from spherical harmonics to many cases of interest by taking linear combinations likewise, dense subspaces are useful because mathematicians know how to extrapolate from dense subspaces to the desired spaces. 

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... 

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