where r is the position of the active electron, Rs is the radius of the sphere, here chosen to be 100 au, A is the thickness of the radial potential that confines the active charge to the sphere (A = 1 au) and we choose y to be 100. The mass and charge of the electron have been set to unity as has Planck's constant divided by 2.n. The motion of this Hamiltonian system is determined by solving the Hamilton's equations for the active electron.

The Poincare surface-of-section technique is an extension of the WKB approximation for non-separable systems in higher dimensions that has the virtue of yielding exact semiclassical results. It has been shown that this technique can be used to determine, semiclassically, the energy levels of a Hamiltonian system which exhibits quasi-periodic behavior. We use the case of three or four excess electrons for illustrative purposes in this contribution. The Poincare surface-of-section technique has been presented elsewhere.23


We have applied this semiclassical quantization scheme for the case of three electrons located on the sphere. Minimum energy conditions dictates that the initial configuration is composed of the three electrons located at the vertices of an equilateral triangle. We have chosen that the plane of the equilateral triangle should correspond with the x-z plane of our calculation. The two fixed electrons are placed at the distance Rs from the center of the sphere and at their appropriate vertex positions if the active electron where set at the position (0, 0, Rs) at the "north pole" of the sphere. The active electron is then give an initial momentum in the x direction while being initially displaced in the y direction. The classical trajectories are calculated for these initial conditions for a number of energies and the resulting table of actions and energies are analyzed to find the best fit to the action quantization conditions. Table 1 shows the result of these calculations as a table of eigenenergies for the case of three electrons on a sphere. Figure 2 shows electron trajectories on a quantum drop with four excess electrons, the electron orbits are shown in red and the polymer nanoparticle is shown in blue.

Table 1. The four lowest energy levels for a quantum drop with three electrons.


Energy (au)

0 0

Post a comment