Laplacian operator, spherical coordinates

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... 

In addition to these new two functions, the Laplacian operator V2 is written in spherical coordinates as... 

The wave function, V /(r), is a function of the vector position variable r. To determine it at every point in space it is convenient to take advantage of the fact that the potential V(r) depends only on the scalar interatomic distance r. In spherical coordinates (Figure 1.2), the Laplacian operator V2 has the form... 

Since the potential energy is spherically symmetrical (a function of r alone), it is obviously advantageous to treat this problem in spherical polar coordinates r, 6, . Expressing the Laplacian operator in these coordinates [cf. Eq (6.21)],... 

By virtue of the presumed spherical syimnetry of this problem, we carry out a separation of variables, with the result that the radial and angular terms are decoupled. This may be seen by starting with the Laplacian operator in spherical coordinates which leads to a Schrodinger equation of the form. 

This form of the equation is not easily applied to rotational motion because the Cartesian coordinates used do not reflect the centro-symmetric nature of the problem. It is better to express the Schrodinger equation in terms of the spherical polar coordinates r, 6 and 0, which are shown in Figure 5.7. Their mathematical relationship to x. y and z is given on the left of the diagram. In terms of these coordinates the Laplacian operator becomes ... 

We can proceed in this way to obtain higher order derivatives, but only first and second derivatives are needed to handle most engineering problems. For cylindrical and spherical coordinates, the shifted position procedure in Problem 12.8 shows how to deal with the Laplacian operators. 

A rigid rotator is a system of two pointlike masses, mi and m2, with a constant distance R between them. The Schrodinger equation may be easily separated into two equations, one for the center of mass motion and the other for the relative motion of the two masses (see Appendix 1 available at booksite.elsevier.com/978-0-444-59436-5 on p. e93). We are interested only in the second equation, which describes the motion of a particle of mass /r equal to the reduced mass of the two particles, and the position in space given by the spherical coordinates R, 9, (/>, where 0kinetic energy operator is equal to - A, where the Laplacian A represented in the spherical coordinates is given in Appendix H available at booksite.elsevier.com/978-0-444-59436-5 on p. e91. Since is a constant, the part of the Laplacian that depends on the differentiation with respect to R is absent. In this way, we obtain the equation (equivalent to the Schrodinger equation) for the motion of a particle on a sphere ... 

The validity of Eq. (4.110) can also be shown by transforming p, i.e., the Laplacian A, given in Cartesian coordinates to spherical coordinates. It then turns out that the components of the angular momentum operator read... 

The coordinate conversion doesn t affect the physics that represents. We can see in Eq. 3.3 that the Laplacian is still a sum over three second derivatives, and each term still has units of (distance) . Combining Eqs. 3.2 and 3.3, we obtain the kinetic energy operator in spherical coordinates ... 

The Laplacian operator in spherical polar coordinates becomes... 

The spherical polar expression for the Laplacian operator appears much more foreboding than the Cartesian coordinate version. However, this is not really the case, since now the Schrddinger equation can be split into radial and angular equations that can be solved separately. To see this, we first write the wavefunction as a product of a function that only depends on r and a function that only depends on the angles 9 and < ... 

In this equation, the first term gives the kinetic energy Xp of an electron in the pth orbital using the Laplacian differential operator V. The Laplacian operator is written out in full for the spherical polar coordinate system in Equation (A9.5). 

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... 

This operator is related to the Laplacian operator, (del squared), in spherical polar coordinates. From the definition of in Cartesian coordinates, and from the coordinate transformation given previously. 

Here, ma is the mass of the nucleus a, Zae2 is its charge, and Va2 is the Laplacian with respect to the three cartesian coordinates of this nucleus (this operator Va2 is given in spherical polar coordinates in Appendix A) rj a is the distance between the jth electron and the a1 1 nucleus, rj k is the distance between the j and k electrons, me is the electron s mass, and Ra>b is the distance from nucleus a to nucleus b. 

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