Spherically symmetric problems

Here S Nft = 0 because of the electroneutrality condition.) Equation (1.3.10) is substituted into Eq. (1.3.6) and the Laplace operator is expressed in polar coordinates (for the spherically symmetric problem) ... 

It is convenient to define a new Thiele-type modulus 4>s for this spherically symmetric problem as ... 

The construction of spherical harmonics can be extended to other dimensions. For example, V. Fock uses four-dimensional spherical harmonics in his article on the S O (4) symmetry of the hydrogen atom — see Chapter 9. Spherical harmonic functions of various dimensions are used in many spherically symmetric problems in physics. 

In this section we have verified mathematically what physicists have tested with long use. In spherically symmetric problems in L ( R ), the spherical harmonics of various degrees are the sensible building blocks they leave nothing out (Proposition 7.5) and they have no substitutes (Proposition 7.6). 

The following proposition justifies the rehance on spherical harmonics in spherically symmetric problems involving the Laplacian. To state it succinctly, we introduce the vector space C2 C 2(R3) continuous functions whose first and second partial derivatives are all continuous. 

Using this potential the molecular many-centre problem is changed to a spherically symmetrical problem such as for atoms as visualized for an eight-atom cluster in Fig. 1. 

Due to the spherical symmetrical problem, an spherical symmetrical equilibriiun equation can be written as ... 

Another example of the use of this statement in boundary value problems is given by the wavefunction i> (r) = Ce-r for a hydrogen atom in a sphere. It is clear that the constant energy value E(R) = -1/2 au for any R value corresponds to the ground state of the spherically symmetric problem with the boundary condition 3 Vr = — f [29]. 

A do not remove xjr from Dr- Hence, one may use angular momentum conservation for spherically symmetric problems. 

There are a number of similar situations where the methods described here may be applied. To be more specific, let us consider some parameter rj and the family of spherically symmetric problems for potentials V(r, rj), Hamiltonians Hv, energies E(rj) and corresponding normalized radial wavefunctions tp(r, rj) with the boundary condition cp(0, rj) = 0 at the center of a sphere of radius R. It follows from the Schrodinger equation that for the derivative dvcp of the wavefunction [Pg.58]

As was shown at the end of Section 6.1, for the Dirichlet problem, G > 0. Hence for any monotonically decreasing /(/ ), the mean value (/) decreases with increasing R. This statement does not depend on the sign of the monotonic function, as was noted in [98] for the ls-Dirichlet problem. This statement holds for the lowest state with any given angular momentum in a spherically symmetric problem. For example, the monotonically decreasing r 3) data for the Dirichlet problem are presented in [101] for 2p states in the sphere of enlarged radius R. 

In order to investigate spherically symmetric problems, it is adequate to introduce polar coordinates (r, r), ip) which are related to the cartesian coordinates of the same point by... 

In this form the parametric solutions, Eqs. (73) and (74), for the spherically symmetric problem can be applied immediately. Physically this implies that a bubble growing in an initially arbitrary temperature field grows at precisely the same rate as if the initial temperature were averaged in each thin spherical shell surrounding the bubble center. Two special cases are considered in detail (7) the linear thermal boundary layer, of thickness /, next to the heating surface, outside of which the temperature is uniform and (2) the exponential boundary layer, where the temperature is assumed to decay exponentially with distance from the wall. The latter distribution is of the form... 

One model equation whose solution has proved useful in many cases is the constant collision frequency, relaxation model of the Boltzmann equation which, for spherically symmetric problems, in spherical coordinates is... 

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