Discrete Spectrum Model

For classical evolutions, we merely substitute crj for p. Looking at plots of N fi, p vs. v/N, it is clear that although the quantum dynamics generally appears to preserve the characteristic structure of the classical spectrum, particular structural details tend to be washed-away [ilachSSbj. If high or low frequency components are heavily favored in the classical evolution, for example, they will similarly be favored in the quantum model discrete peaks, however, will usually disappear. White-noise spectra, of course, will remain so in the quantum model. 

IR instruments are available in filter-based, grating-based, and FT-based models. The usual approach is to use a full-spectrum model to ascertain the working wavelengths for a particular reaction, then to apply simpler filter instruments to the process. This works where one, two, or three discrete wavelengths may be used for the analysis. If complex, chemometric models are used, and full-spectrum instruments are needed. 

The concept can be illustrated with a simple one-dimensional problem, in which a spherically symmetric potential well is surrounded by a barrier (see Fig. 4). One might consider this problem as a model for dissociation of a diatomic molecule. According to the general theory, the spectrum is discrete for < 0 and continuous for E > 0. Let us now look for unbound solutions, x( ) = of fhe time-independent Schrodinger equation which behave like exp(ifci ) at large R. The desired wave functions have the form... 

The problem will be solved for the case where the viscoelastic half-plane is characterized by a discrete spectrum model (Sect. 1.6). The more general continuous spectrum model is discussed by Golden (1977). The proportionality assumption (Sect. 1.9) will be adopted for the material so that a unique Poisson s ratio exists. Therefore, from (1.6.25, 28, 29), (3.5.20) and (3.5.22), we have... 

It should be noted that (3.7.23) is essentially as general as an arbitrary discrete or continuous spectrum model, in the present context. This is because all quantities must be linear in the viscoelastic functions, so that the results for a more general model are simply sums of terms of the form that will now be derived. Explicit results can be obtained for the problem with friction, in terms of Whittaker functions. However, these will not be introduced in the present work. We refer to Golden (1979a, 1986a) for further details. In the frictionless case (from (3.7.11) we see that d,o = 0) ... 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

Here, W is a cut-off of the order of the 7t-band width, introduced because the right-hand side of Eq. (3.13) is formally divergent. As in the discrete model, the spectrum of eigenstates of Hct for A(a)= Au has a gap between -Ao and +Alh separating the empty conduction band from the completely filled valence band. 

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

The four-coordinate complexes [Mo(SBu,)4] and [Mo(NR2)4] are also diamagnetic. The UV-PE spectrum of [Mo(SBu )4] exhibits a low ionization potential at 6.8 eV which has been assigned to the ionization of electrons with predominant molybdenum Adz2 character, on the basis of discrete variational-Xa MO calculations on the model compounds [Mo(SH)4] and [Mo(SMe)4].210 For [Mo(NR2)4] (R = Me, Et), the UV-PE spectrum contains a low energy ionization at 5.3 eV which has been attributed to ionization from the molybdenum Adx2-y orbital. This assignment was based on Fenske-Hall calculations on [Mo(NMe2)4].2U... 

The relaxation time distribution of the bead-spring models is discrete. The spectrum is... 

Tn the Rohr model of the hydrogen atom, the proton is a massive positive point charge about which the electron moves. By placing quantum mechanical conditions upon an otherwise classical planetary motion of the electron, Bohr explained the lines observed in optical spectra as transitions between discrete quantum mechanical energy states. Except for hvperfine splitting, which is a minute decomposition of spectrum lines into a group of closely spaced lines, the proton plays a passive role in the mechanics of the hydrogen atom, It simply provides the attractive central force field for the electron,... 

Now that we ve seen how atomic structure is described according to the quantum mechanical model, let s return briefly to the subject of atomic line spectra first mentioned in Section 5.3. How does the quantum mechanical model account for the discrete wavelengths of light found in a line spectrum ... 

Nano-scale and molecular-scale systems are naturally described by discrete-level models, for example eigenstates of quantum dots, molecular orbitals, or atomic orbitals. But the leads are very large (infinite) and have a continuous energy spectrum. To include the lead effects systematically, it is reasonable to start from the discrete-level representation for the whole system. It can be made by the tight-binding (TB) model, which was proposed to describe quantum systems in which the localized electronic states play an essential role, it is widely used as an alternative to the plane wave description of electrons in solids, and also as a method to calculate the electronic structure of molecules in quantum chemistry. 

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