Spherical polar coordinates transform

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 transformation between cartesian and spherical polar coordinates is... 

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 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... 

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

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]. 

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

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... 

When the cartesian coordinates are transformed into, spherical polar coordinates and normalization is taken into consideration, the three forms of p-orbitals take the following expressions ... 

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. 

The classical energy of the system can be expressed in Cartesian coordinates and is simply transformed into spherical polar coordinates. 

As you have seen, the Schrodinger equation may be written in spherical polar coordinates using the usual transformation. As a result, we can write a radial part and an angular part for the wavefunction ik. As an example, let s look at the wavefunction for the Is and 2pz orbitals in the hydrogen atom. These orbitals have the quantum numbers n = 1, l = 0, mg = 0 and n = 2, = 1, mg = 0, respectively. 

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... 

The usual spherical polar coordinate system (r, 6, right-handed Cartesian coordinate system whose coordinates occur in the equations. The Z-axis is the unique axis around which r-functions (A = 0) have rotational symmetry. Functions with 0 < A g / are characterized 19) through A by their standard transformation properties when they undergo rotations around the Z-axis ... 

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

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

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

A triple integral in Cartesian coordinates is transformed into a triple integral in spherical polar coordinates by... 

For instance, if we are concerned with a diatomic molecule, it is convenient to set the bond initially along the Z axis. Thus, X = (0, 0, re), where re is the equilibrium bond length. After the transformation, the space-fixed coordinates are re (sin 0 cos 4>, sin 0 sin 4>, cos 0), corresponding to the usual spherical polar coordinates. 

---