Particle-on-a-sphere problem

The first term is called the radial wavefunction, and describes the in-out motion of the electron. It is written with the subscripts to show that the mathematical form of the function depends on the values of n and l we shall look at some examples below. The second term in eqn 4.18 is exactly the same spherical harmonic that arises in the particle-on-a-sphere problem. It... 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

This allows for the equivalence between crossed cylinders and the particle on a plane problem. Likewise, the mechanics of two spheres can be described by an equivalently radiused particle-on-a-plane problem. The combination of moduli and the use of an effective radius greatly simplifies the computational representation and allows all the cases to be represented by the same formula. On the other hand, it opens the possibility of factors of two errors if the formula are used without realizing that such combinations have been made. Readers are cautioned to be aware of these issues in the formulae that follow. 

In this fashion the linear combinations are assigned quantum numbers 1 and m which are related to those which have been defined for the S, P and D functions derived for the particle on the sphere problem. Furthermore, their nodal characteristics mimic those of the atomic wave functions of the central atom. The spherical harmonic expansion described above will provide its most accurate description of the symmetry adapted linear combinations when the polyhedral vertices are symmetry equivalent. For example, octahedral MLg has S°, Po, i and Do,2s a total of 6 symmetry adapted linear combinations. They are... 

A richer zoo of magic numbers can be obtained by filling the sphere. Then the qualitative types (a)-(d) in Figure 15.1 can be imagined. Case (a) is that of the thin attractive shell or particle on a sphere. Case (b) is a wine bottle potential, with a repulsive excess at the centre. Case (c) is the classic problem of a particle in a sphere. Case (d) is a parabolic one, possibly flattened at the bottom. 

Problem (a) Confirm that Ym is an eigenfunction of the Particle-on-a-Sphere Hamiltonian, and (b) that it is normalized. 

Confirm that Ylo and Yi i as given in Table 3-1 are a) eigenfunctions of the Particle-on-a-Sphere model problem b) normalized and c) orthogonal. 

Because the radial and angular parts are separable and the molecule rotates freely in space, the angular part of Equation 6-6 is identical to the Particle-on-a-Sphere model problem developed in Section 3.2. Hence, the angular functions Yim are the spherical harmonics (Equation 3-19). The solution of the A operator applied to Y is known and given in Equation 3-20. 

Closely related to the problem of a particle on a line is that of a particle confined to a hollow sphere. Such a particle is described by the same Hamiltonian as a free particle (V = 0), i.e. 

Stone applied similar reasoning to the problem of a three-dimensional cluster. Here, the solutions of the corresponding free-particle problem for an electron-on-a-sphere are spherical harmonics. These functions should be familiar because they also describe the angular properties of atomic orbitals. Two quantum numbers, L and M, are associated with the spherical harmonics, [Pg.1219]

Therefore, solving the problem of a reduced mass 11 rotating about a fixed point at the fixed distance r = ri -h r2 is equivalent to solving the two-mass rigid-rotor problem. In effect, the rotating-diatomic problem is transformed to a particle-on-the-surface-of-a-sphere problem. 

I he field scattered by any spherically symmetrical particle composed of materials described by the constitutive relations (2.7)-(2.9) has the same form as that scattered by the homogeneous sphere considered in Chapter 4. However, the functional form of the coefficients an and bn depends on the radial variation of e and ju. In this section we consider the problem of scattering by a homogeneous sphere coated with a homogeneous layer of uniform thickness, the solution to which was first obtained by Aden and Kerker (1951). This is one of the simplest examples of a particle with a spatially variable refractive index, and it can readily be generalized to a multilayered sphere. 

Let us apply the interpolation procedure to a case involving an electric field. It is well known that the efficiency of the granular bed filters can be significantly increased by applying an external electrostatic field across the filter. In this case, fine (<0.5-/rm) particles deposit on the surface of the bed because of Brownian motion as well as because of the electrostatically generated dust particle drift [51], The rate of deposition can be calculated easily for a laminar flow over a sphere in the absence of the electrostatic field [5]. The other limiting case, in which the motion of the particles is exclusively due to the electric field, could also be treated [52], When, however, the two effects act simultaneously, only numerical solutions to the problem could be obtained [51],... 

Here the reaction rate depends not only on the concentration of the S02 but also on the surface area of any catalyst available, such as airborne dust particles. The efficiency of a catalyst depends upon its specific surface area, Asp, defined as the ratio of surface area to mass [30], Accordingly, this property is frequently used as a basis for comparing different kinds of catalysts, or catalyst supports, and for diagnosing practical problems in catalysts being used in a process (since both agglomeration and poisoning reduce Asp). The specific surface area of the airborne dust particles, considering n spheres of density p and radius R, would be ... 

Widom [9] realized the importance of this problem for statistical mechanics and showed that the centers of the particles of a hard disk gas, in an equilibrium position, are not uniformly random distributed. The available area for a nevt particle power series in particle density 6 = Nnr2/A, where N is the number of adsorbed panicles, r their radius and A the total area of the surface. The coefficients of the series terms are identical up to the second power of 9 for the equilibrium and the RSA models. The differences in the higher powers coefficients lead for RSA to jamming for Op = 0.76, 0.547 and 0.38 for the ID (segments on a line), 2D (disks on a surface) and 3D (spheres on a volume), respectively, while for the equilibrium configurations the close-packing occurs at 9 = 1, 0.91 and 0.74, respectively. 

The hierarchy of equations thereby obtained can be closed by truncating the system at some arbitrary level of approximation. The results eventually obtained by various authors depend on the implicit or explicit hypotheses made in effecting this closure—a clearly unsatisfactory state of affairs. Most contributions in this context aim at calculating the permeability (or, equivalently, the drag) of a porous medium composed of a random array of spheres. The earliest contribution here is due to Brinkman (1947), who empirically added a Darcy term to the Stokes equation in an attempt to represent the hydrodynamic effects of the porous medium. The so-called Brinkman equation thereby obtained was used to calculate the drag exerted on one sphere of the array, as if it were embedded in the porous medium continuum. Tam (1969) considered the same problem, treating the particles as point forces he further assumed, in essence, that the RHS of Eq. (5.2a) was proportional to the average velocity and hence was of the explicit form... 

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