Energy spectrum bounded systems

The particle density in an isolated bounded system is required to be zero at the boundary point at infinity. This introduces gaps (or discreteness) in the excitation spectrum, at low energy, which are not present in extended systems. The presence of shell structure, and whether the shells are open or closed, is... 

Even in the framework of nonrelativistic quantum mechanics one can achieve a much better description of the hydrogen spectrum by taking into account the finite mass of the Coulomb center. Due to the nonrelativistic nature of the bound system under consideration, finiteness of the nucleus mass leads to substitution of the reduced mass instead of the electron mass in the formulae above. The finiteness of the nucleus mass introduces the largest energy scale in the bound system problem - the heavy particle mass. 

Bounded quantum systems have in general a discrete energy spectrum formed by a sequence of energy eigenvalues . Discrete vibrational spec-... 

In summary, we have shown how the absorption of a photon leads to the formation of a resonant scattering state. Explicit formulas involving quadrature over the system energy spectrum have been presented but not evaluated. When the resonant scattering state may be approximated in terms of a set of quasistationary bound states, an explicit relationship is obtained for the rate of dissociation in terms of the matrix elements coupling zero-order states and the corresponding densities of states. In principle this permits the use of experimental rate data to evaluate the matrix elements vx and v2, if px and p2 can be estimated. 

Systems in the collinear eZe configuration which have tori would be the antiproton-proton-antiproton (p-p-p) system, the positronium negative ion (Pr- e-e-e)), which corresponds to the case of Z= 1, = 1, and If these systems have bound states, we can see the effect of our finding in the Fourier transform of the density of states for the spectrum. For a positronium negative ion, the EBK quantization was done [34]. Stable antisymmetric orbits were obtained and were quantized to explain some part of the energy spectrum. As hyperbolic systems, H and He have been already analyzed in Refs. 11 and 17, respectively. Thus, Li+ is the next candidate. We might see the effect of the intermittency for this system in quantum defect as shown for helium [14]. 

Notice something very important about these results. The application of the boundary conditions has led to a series of quantized energy levels. That is, only certain energies are allowed for the particle bound in the box. This result fits very nicely with the experimental evidence, such as the hydrogen emission spectrum, that nature does not allow continuous energy levels for bound systems, as classical physics had led us to expect. Note that the energies are quantized, because the boundary conditions require that n assume only integer values. Consequently, we call n the quantum number for this system. 

A quantum dot is made from a semiconductor nanostructure that confines the motion of conduction band electrons, valence band holes, or excitons (bound pairs of conduction band electrons and valence band holes) in all three spatial directions. A quantum dot contains a small finite number (of the order of 1 to 100) of conduction band electrons, valence band holes, or excitons, that is, a finite number of elementary electric charges (Scheme 16.2). The reason for the confinement is either the presence of an interface between different semiconductor materials (e.g. in coie-sheU nanocrystal systems) or the existence of the semiconductor surface (e.g. semiconductor nanocrystal). Therefore, one quantum dot or numerous quantum dots of exactly the same size and shape have a discrete quantized energy spectrum. The corresponding wave functions are spatially localized within the quantum dot, but they always extend over many periods of the crystal lattice (5). 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

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