Transformation to spherical polar coordinates

Equation (12) can be solved analytically, by separating the variables according to standard procedures. Because of the nature of the Coulomb potential it is necessary to transform to spherical polar coordinates first i.e. 

The relative Schrodinger equation cannot be solved in Cartesian coordinates. We transform to spherical polar coordinates in order to have an expression for the potential energy that contains only one coordinate. Spherical polar coordinates are depicted in Figure 17.3. The expression for the Laplacian operator in spherical polar coordinates is found in Eq. (B-47) of Appendix B. The relative Schrodinger equation is now... 

The transformation to spherical polar coordinates makes the potential energy dependent on only the variable, r, rather than on all six coordinates. The kinetic energy has a contribution associated with translational motion of the whole system. 

Taking the proton to be fixed in space makes Zp a constant. So, the interaction with the proton is a constant contribution to the energies of all the states we can ignore its effect. The coordinate z should be transformed to spherical polar coordinates since they were used to solve for the wavefunctions of the unperturbed hydrogen atom. 

Transformation from Cartesian coordinates xyz to spherical polar coordinates r 6 ip is a standard exercise, which yields... 

It turns out that the solutions of (6.5) are much simpler if one transforms from cartesian to spherical polar coordinates, as defined in figure 6.1. The relationships between the two are... 

Figure 20-1. Transformation from Cartesian to spherical polar coordinates...

The following equations and Eq. (2.58) can be used to transform from Cartesian coordinates to spherical polar coordinates ... 

If we substitute Eq. (25-30) and the fields of Table 25-2 into Eq. (25-3), the dipole fields are given exactly by integration over two modes for z 0 and z 0. Asymptotic evaluation of these integrals for r - c , together with a transformation from cylindrical to spherical polar coordinates, leads to the far fields of Eqs. (21-23) and (21-24a). 

Transform the components of angular momentum Mx, My, and M, found in Probleni 12, to spherical polar coordinates. 

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... 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

These functions are expressed in terms of spherical polar coordinates (r,0,) but, to evaluate the overlap integrals it is easier to transform to spheroidal coordinates ( , ,< ). The two sets of coordinates are related by the expressions... 

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... 

If one uses the chain rule to transform the Laplacian V2= d2/dx2 + 92/ 9y2+ 92/9z2 into spherical polar coordinates, the result is... 

Figure 1.3. Transformation from space-fixed axes X, Y, Z to molecule-fixed axes using the spherical polar coordinates R,Q,

Intuitively an electron would not be expected to move parallel to the orthogonal x, y and z axes. In fact, it makes more sense to describe the wavefunction in terms of the spherical polar coordinates r, 9 and . The first stage in analysing the hydrogen atom in quantum mechanics is therefore to transform T (x, y, z) into 9, (p). This is quite an involved process mathematically. 

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