Wave equation in spherical polar coordinates

Spherical Bessel Functions. A problem which arises in mathematical physics is that of the solution of the wave equation in spherical polar coordinates... 

The choice of generating functions is dictated by whatever symmetry may exist in the problem. In this chapter we are interested in scattering by a sphere therefore, we choose functions ip that satisfy the wave equation in spherical polar coordinates r, 6,

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The scalar wave equation in spherical polar coordinates is... 

This equation can be solved by separation of variables, provided the potential is either a constant or a pure radial function, which requires that the Lapla-cian operator be specified in spherical polar coordinates. This transformation and solution of Laplace s equation, V2 / = 0, are well-known mathematical procedures, closely followed in solution of the wave equation. The details will not be repeated here, but serious students of quantum theory should familiarize themselves with the procedures [15]. 

Solving this equation will not concern us, although it is useful to note that it is advantageous to work in spherical polar coordinates (Figure 1.4). When we look at the results obtained from the Schrodinger wave equation, we talk in terms of the radial and angular parts of the wavefunction,... 

For systems of chemical interest the amplitude function ip that occurs as a solution of (4.19) is postulated to give a complete description, provided the potential energy V, is correctly specified. In reality, the only chemically significant problem that has been solved is of an electron associated with an isolated stationary proton, with potential energy V = jr, in atomic units. The differential wave equation is separable in spherical polar coordinates. Separate solutions, as functions of radial (r) and angular 9, ip) coordinates, describe the quantized energy and angular momentum of the electron as ... 

The kinetic energy operator,however,is almost separable in spherical polar coordinates, and the actual method of solving the differential equation can be found in a number of textbooks. The bound solutions (negative total energy) are called orbitals and can be classified in terms of three quantum numbers, n, I and m, corresponding to the three spatial variables r, d and q>. The quantum numbers arise from the boundary conditions on the wave function, i.e. it must be periodic in the 0 and q> variables, and must decay to zero as r oo. Since the Schrodinger equation is not completely separable in spherical polar coordinates, there exist the restrictions n > /> m. The n quantum number describes the size of the orbital, the / quantum number describes the shape of the orbital, while the m quantum number describes the orientation of the orbital relative to a fixed coordinate system. The / quantum number translates into names for the orbitals ... 

Since the interaction (4.304) is central, the associate wave equation may be separated in spherical polar coordinates to produce the normalized radial function. For the bound states hydrogenic atoms in the case of an infinitely heavy nucleus it looks like (Bransden Joachain, 1983) ... 

An angular wave function, Y 9, 4>), is the part of a wave function that depends on the angles 9 and when the Schriidinger wave equation is expressed in spherical polar coordinates. (See also radial wave function.)... 

In Chapter I we found that curvilinear coordinates, such as spherical polar coordinates, are more suitable than Cartesian coordinates for the solution of many problems of classical mechanics. In the applications of wave mechanics, also, it is very frequently necessary to use different kinds of coordinates. In Sections 13 and 15 we have discussed two different systems, the free particle and the three-dimensional harmonic oscillator, whose wave equations are separable in Cartesian coordinates. Most problems cannot be treated in this manner, however, since it is usually found to be impossible to separate the equation into three parts, each of which is a function of one Cartesian coordinate only. In such cases there may exist other coordinate systems in terms of which the wave equation is separable, so that by first transforming the differential equation into the proper... 

Wave functions are most easily analyzed in terms of the three variables required to define a point with respect to the nucleus. In the usual Cartesian coordinate system, these three variables are the x, y, and z dimensions. In the spherical polar coordinate system, they are r, the distance of the point from the nucleus, and the angles 6 (theta) and 0 (phi), which describe the orientation of the distance line, r, with respect to the x, y, and z axes (Fig. 8-21). Either coordinate system could be used in solving the Schrodinger equation. 

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),... 

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