Eigenfunctions spherical coordinates

The Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

In principle, knowledge of Eqs. [18]—[22] is sufficient to set up differential equations for the orbital angular momentum operators and to solve for eigenvalues and eigenfunctions. The solutions are most easily obtained employing spherical coordinates r, 0,< ) (see Figure 7). The solutions, called spherical harmonics, can be found in any introductory textbook of quantum chemistry and shall be given here only for the sake of clarity. 

If we pass to polynomials of degree 2, we have to be careful. They do not always look homogeneous when we finish setting x2 + y2 + z2 = 1. Let us look at z2, for example, which is not harmonic and so should not be an eigenfunction of A. Passing to spherical coordinates, and setting r = 1, this gives the function... 

E. AXISYMMETRIC CREEPING FLOWS SOLUTION BY MEANS OF EIGENFUNCTION EXPANSIONS IN SPHERICAL COORDINATES (SEPARATION OF VARIABLES)... 

Again, it is emphasized that though the form of this solution does depend on the fact that we have used eigenfunctions for a spherical coordinate system, it is strictly independent of the body geometry this will come into play only when we try to apply boundary conditions at the body surface to determine C and D . For now we simply leave these constants unspecified. 

In summary then, the leading-order problem is just the translation of a spherical drop through a quiescent fluid. The solution of this problem is straightforward and can again be approached by means of the eigenfunction expansion for the Stokes equations in spherical coordinates that was used in section F to solve Stokes problem. Because the flow both inside and outside the drop will be axisymmetric, we can employ the equations of motion and continuity, (7-198) and (7-199), in terms of the streamfunctions f<(>> and < l)), that is,... 

Determine the velocity and pressure fields in the liquid as well as the velocity of the bubble by means of a full eigenfunction expansion for spherical coordinates. Does this solution satisfy the normal-stress condition on the bubble surface ... 

Problem 7-20. Sphere in a Parabolic Flow. Use the general eigenfunction expansion for axisymmetric creeping-flows, in spherical coordinates, to determine the velocity and pressure fields for a sohd sphere of radius a that is held fixed at the central axis of symmetry of an unbounded parabolic velocity field,... 

The same eigenfunctions in spherical coordinates used in the previous subsection and introduced in 3.1 are the basis for the analysis of the hydrogen atom confined by a circular cone defined by a fixed value of the polar angle 9 = 9q. The boundary condition requiring the vanishing of the wave function at such an angle must be satisfied by the hypergeometric function in Equation (36),... 

The S eigenfunctions (in spherical coordinates) and their energy levels in a charged particle (electron or hole) with effective mass m confined in a spherical well of infinite depth and radius are defined in classical quantum mechanics by the equations... 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

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. 

Hence there exists a complete set of common eigenfunctions for L2 and any one of its components. The eigenvalue equations for L2 and Lz are found to be separable in spherical polar coordinates (but not in Cartesian coordinates). Using the chain rule to transform the derivatives, we can find... 

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... 

The components may be expressed in either a space-fixed axis system (p) ora molecule-fixed system (q). The early literature used cartesian coordinate systems, but for the past fifty years spherical tensors have become increasingly common. They have many advantages, chief of which is that they make maximum use of molecular symmetry. As we shall see, the rotational eigenfunctions are essentially spherical harmonics we will also find that transformations between space- and molecule-fixed axes systems, which arise when external fields are involved, are very much simpler using rotation matrices rather than direction cosines involving cartesian components. 

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