Laplacian spherical coordinates

To obtain the Laplacian in spherical coordinates appropriate second derivatives. Again, as an exan (16) can be written as... 

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... 

In our discussion of spherical harmonics we will use an expression of the three-dimensional Laplacian in spherical coordinates. For this we need spherical coordinates not just on but on all of three-space. The third coordinate is r, the distance of a point from the origin. We have, for arbitrary (x, y, zY e... 

Exercise 3.26 (Used in Proposition A.3) Consider the Laplacian in spherical coordinates (see Exercise 1.12) ... 

If we consider spherical vessels of radius ro such that T will depend only on the distance from the center r, then T = T r,t)y and the Laplacian " can be written in spherical coordinates... 

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. 

Hence, the radial contribution to the spherical-coordinate Laplacian of molar... 

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 / is absent In this way, we obtain the equation (equivalent to the Schrddinger equation) for the motion of a particle on a sphere ... 

Laplacian in spherical coordinates (Appendix H available at booksite.elsevier.com/978-0-444-59436-5, p. e91, recommended)... 

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. 

After inserting the Laplacian (in spherical coordinates, see Appendix H on p. 969) and the product (6.20) into (6.18) we obtain the following series of transformations... 

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... 

A Verify Eq. (6.6) for the Laplacian in spherical coordinates. (This is a long, tedious problem, and you probably have better things to spend your time on.)... 

Table 1 7.1 For a scalar / or a vector u, this table gives the gradient cif, the divergence V u, and the Laplacian V-/ in Cartesian, cylindrical, and spherical coordinates. 

We have to make one major adjustment before proceeding we have to figure out the best way to write the Laplacian, V. For atomic systems, spherical coordinates are much more convenient to use than Cartesian coordinates. 

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 ... 

In spherical coordinates and spherical symmetry, the Laplacian contains only the radial term, which is ... 

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... 

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