Laplacian in spherical coordinates

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

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) ... 

After inserting the Laplacian (in spherical coordinates see Appendix H available at booksite. elsevier.com/978-0-444-59436-5 on p. e91) and the product [Eq. (6.20)] into Eq. (6.18), we obtain the following series of transformations ... 

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

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.)... 

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... 

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. 

EXAMPLE 7.29 Write the expression for the Laplacian in spherical polar 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... 

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. 

X = r sin0 cos< , Y = r sin0sin0, Z = rcos9, so that, in spherical polar coordinates, the Laplacian is given by... 

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)],... 

Notice finally that the appearance of the logarithm in the inflection point criterion is related to ln(r) being the two-dimensional Coulomb potential, i.e., the Green function of the cylindrically symmetric Laplacian. In the corresponding three-dimensional (spherical) problem of charged colloids the Green function 1/r would be the appropriate choice for plotting the radial coordinate [31,32], More details can be found in Ref. 4. 

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