Discrete sets of energy levels

These factors, in turn, are dependent on the diameter and helicity. It has been found that metallicity occurs whenever (2n + m) or (2 + 2m) is an integer multiple of three. Hence, the armchair nanotube is metallic. Metallicity can only be exactly reached in the armchair nanotube. The zigzag nanombes can be semimetallic or semiconducting with a narrow band gap that is approximately inversely proportional to the tube radius, typically between 0.5 -1.0 eV. As the diameter of the nanombe increases, the band gap tends to zero, as in graphene. It should be pointed out that, theoretically, if sufficiently short nanotubes electrons are predicted to be confined to a discrete set of energy levels along all three orthogonal directions. Such nanotubes could be classified as zero-dimensional quantum dots. 

Currently, quantum mechanics provides the most complete theoretical understanding of spectroscopy and the information that spectroscopic analysis yields. Quantum theory predicts a discrete set of energy levels for particles, and, therefore, the reflection, transmission and absorption characteristics of a sample can be compared to the characteristics of known materials over a spectrum of wavelengths, thus providing a means of identification of a sample. Since spectroscopy could, conceivably, cover a vast number of methods of analysis, the particular kinds of information that can be acquired through the use of spectroscopic analysis is best illustrated by way of several examples. Astronomical spectroscopy... 

When one solves for the possible energies for the particles in a potential U, the result is that you will have a continuous energy spectrum of free particles above the potential, and a discrete set of energy levels in the potential, see Fig.(1.1). The discrete set is called bound states and represent the energies of the particles, usually electrons, that are bound by the potential. This does not mean that the particles do not move, of course, only that they have a fix energy. 

For the numerical evaluation, Eq. (21.58) can be approximated by replacing the integration with a discrete sum over a set of energy levels. The result of such an evaluation is displayed in Fig. 21.5(d), which compares the classical Arrhenius rate with the quantum mechanical rates calculated from Eq. (21.56). 

No, molecules actually have a discrete number of possible energy levels. Another way to say this is that their energies are quantized. To illustrate why this is so different from situations we re used to in everyday life, consider what happens when you re throwing a baseball. You could throw it at any speed between 0 meters per second (m/s) and however fast you are capable of throwing it. In molecules, though, only a discrete set of energies are possible. It s as if you could throw the baseball either 2 m/s or 40 m/s, but not 20 m/s or any other speed in between. There aren t many situations we encounter in everyday life in which the possible energies associated with objects come in a discrete set of values. 

The atomic harmonic oscillator follows the same frequency equation that the classical harmonic oscillator does. The difference is that the classical harmonic oscillator can have any amplitude of oscillation leading to a continuum of energy whereas the quantum harmonic oscillator can have only certain specific amplitudes of oscillation leading to a discrete set of allowed energy levels. 

The energies of the various contributions are quantised , i.e., in a given state the isolated molecule may possess one of a discrete set of values these values are often referred to as energy levels. When a molecule absorbs light, its energy is momentarily increased by an amount equal to that of the photon. The energy is related to the wave length (X) and frequency (v) by the equation ... 

When an atom or molecule is adsorbed on a surface new electronic states are formed due to the bonding to the surface. The nature of the surface chemical bond will determine the properties and reactivity of the adsorbed molecule. In the case of physisorption, the bond is rather weak, of the order of 0.3 eV. The overlap of the wave functions of the molecule and the substrate is rather small and no major change in the electronic structure is usually observed. On the contrary, when the interaction energy is substantially higher, there are rearrangements of the valence levels of the molecule, a process often denoted chemisorption. The discrete molecular orbitals interact with the substrate to produce a new set of electronic levels, which are usually broadened and shifted with respect to the gas phase species. In some cases completely new electronic levels emerge which have no resemblance to the original orbitals of the free molecule. 

The resonance current (of order TT)) is formed by non-compensated partial currents carried by the Andreev levels in the vicinity of the Fermi energy, i.e. for E ay = 0 when D —> 0. Such levels exist only for a discrete set of Zeeman splittings... 

One factor affecting the dielectric strength is the electronic structure of the polymer, and in particular its band gap. In quantum mechanics [29], each electron in a molecule can only occupy one of a discrete set of allowed energy levels. In solids, the overlaps between different repeating units of the material (for example, the repeat units in quasi-one-dimensional systems such as polymer chains [29-31]) cause these discrete energy levels to broaden into bands. The band gap is the energy difference between the top of the valence band and the bottom of the conduction band. (In terms which are equivalent but more familiar to chemists, the band gap is... 

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