Characteristic energy values

For a given system the characteristic energy values Wn may occur only as a set of discrete values, or as a set of values covering a continuous range, or as both. From analogy with spectroscopy it is often said that in these three cases the energy values comprise a discrete spectrum, a continuous spectrum, or both. The way... 

There is one such equation for each set of values of the electronic quantum numbers n, and each of these equations possesses an extensive set of solutions, corresponding to the allowed values of the nuclear quantum numbers v. The values of Wn.y are the characteristic energy values for the entire molecule they depend on the electronic and nuclear quantum numbers n and v. 

Let us now consider a system whose Schrodinger time functions corresponding to the stationary states of the system are k0, i, , kn, . Suppose that we carry out an experiment (the measurement of the values of some dynamical quantities) such as to determine the wave function uniquely. Such an experiment is called a maximal measurement. A maximal measurement for a system with one degree of freedom, such as the one-dimensional harmonic oscillator, might consist in the accurate measurement of the energy the result of the measurement would be one of the characteristic energy values W and the corresponding wave function would then represent the... 

The Schrodinger stationary-state wave functions are probability amplitude functions between the energy and the coordinates of the system. For a system with one degree of freedom, such as a harmonic oscillator, the wave functions pn are the transformation functions x W1) between the coordinate x and the characteristic energy values, and for the hydrogen atom the wave functions d, [Pg.433]

An important advance in atomic spectroscopy was made in 1913 by Niels Bohr. According to the Bohr theory any atom is allowed only certain discrete and characteristic energy values. Absorption or emission of radiation is a result of a transition between two energy levels of the atom. These energy levels were found to be directly proportional to the empirical spectral terms of the atom. Planck showed the following relationship between the energy and frequency of a particular radiation ... 

The characteristic energy values for vibrational to vibrational as well as rotational to rotational energy transfer and their weak dependence on temperature relates to the phenomenon of resonance. Transfer becomes less effective when the difference in energies of the modes involved becomes larger. This follows from so-called surprisal analysis (a statistical method) and is due to decreasing cross sections when mode frequencies v become very different. According to surprisal analysis the rate constant for transitions between states labeled v and v depends Boltzmann-like on the energy difference between them ... 

Only certain energy values ( ) will lead to solutions of this equation. The corresponding values of the wave functions are called Eigen functions or characteristic wave functions. 

The requirement that i/ be periodic in d with the period 2ir leads to the functions known as Mathieu functions.3 These are usually designated by the symbols ceo, seit cei, se2, ce2, etc. The functions and the corresponding characteristic values of a as functions of q have been evaluated by Goldstein.4 The energy values for the five lowest states are shown in Fig. 3. It is seen... 

The energy of the photoelectrons leaving the sample are determined using a Concentric Hemispherical Analyser (CHA), and this gives a spectrum with a series of peaks whose energy values are characteristic of each element. A schematic diagram of a CHA is shown in Figure 2.3. 

When the accelerating voltage reaches a specific value (dependent on the nature of the target material), the electrons from the beam are capable of knocking out core-level electrons from the target material, thus giving rise to core vacancies. These are quickly filled by electrons in upper levels and this results in the emission of X-ray photons of characteristic energies which depend on the... 

Our procedure depends on a new computer program, RAMM (RAndom Molecular Mechanics), which is applicable to any kind of biomolecule. It is described in detail elsewhere (KoS r, T./ Petrak, F. Galova, Z. TvaroSka, I. Carbohvdr. Res.. in Press). Only the basic characteristics of RAMM and its application to conformational analysis of disaccharides are discussed here, concentrating on the effect of the orientations of pendant groups on the energy values at the various < ) and f torsion angles. 

With analytical methods such as x-ray fluorescence (XRF), proton-induced x-ray emission (PIXE), and instrumental neutron activation analysis (INAA), many metals can be simultaneously analyzed without destroying the sample matrix. Of these, XRF and PEXE have good sensitivity and are frequently used to analyze nickel in environmental samples containing low levels of nickel such as rain, snow, and air (Hansson et al. 1988 Landsberger et al. 1983 Schroeder et al. 1987 Wiersema et al. 1984). The Texas Air Control Board, which uses XRF in its network of air monitors, reported a mean minimum detectable value of 6 ng nickel/m (Wiersema et al. 1984). A detection limit of 30 ng/L was obtained using PIXE with a nonselective preconcentration step (Hansson et al. 1988). In these techniques, the sample (e.g., air particulates collected on a filter) is irradiated with a source of x-ray photons or protons. The excited atoms emit their own characteristic energy spectrum, which is detected with an x-ray detector and multichannel analyzer. INAA and neutron activation analysis (NAA) with prior nickel separation and concentration have poor sensitivity and are rarely used (Schroeder et al. 1987 Stoeppler 1984). 

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