Harmonic energy spectrum

The most simple way to accomplish this objective is to correct the external field operator post factum, as was repeatedly done in magnetic resonance theory, e.g. in [39]. Unfortunately this method is inapplicable to systems with an unrestricted energy spectrum. Neither can one use the method utilizing the Landau-Teller formula for an equidistant energy spectrum of the harmonic oscillator. In this simplest case one need correct... 

This result determines the energy spectrum of the harmonic oscillator through... 

As a measure of the shape or the collectivity of the nuclei we use the ratio E4/E2, where E2 and E4 are the excitation energies of the spin 2 and spin 4 states. The value of this quantity is 3 1/3 for a nucleus with a pure rotational spectrum, and 2 for a harmonic vibrational spectrum. 

Table 1 Energy spectrum of the g-defonned, 3-dimensional harmonic oscillator for r-dependent Ttuto and q = er with r — 0.028. The magic numbers, marked, correspond to the energy gaps larger than 0.275, reported in boldface.

Figure 4 The energy spectrum of the -deformed, 3-dimensional harmonic oscillator as a function of the deformation parameter r. As in Nilsson model pictures, one observes level bunching areas at certain values of r, separated by larger energy gaps.

Figure 1. Energy spectrum of a three-dimensional harmonic oscillator.

In this equation, by letting Aj = 2 = one immediately recovers the energy spectrum of the three-dimensional harmonic oscillator (truncated up to -I-1 vibrational levels),... 

This is the entropy of each mode of Irequency to. We need to consider all the modes. In a solid, the total number of modes of vibration consisting of N atoms is 3N 6, giving rise to an energy spectrum. The total harmonic entropy of the solid is obtained by integration of Eq. (19.6) over this spectrum. 

This means that the energy spectrum will be compressed for heavier nuclei. If one uses a square well instead of the harmonic oscillator potential, the oscillator levels split up according to different lvalues (Schiff 1968 Nilsson and Ragnarsson 1995). The harmonic oscillator level system is shown on the left side of Fig. 2.9. The second column presents the intermediate energy spectrum, obtained by interpolation between the harmonic oscillator and square well potential levels. 

Figure 11.2. Energy spectrum of a signal (a) harmonic signal (b) turbulent signal...

For a system with harmonic restoring forces the width of the energy spectrum would increase linear with temperature and vanish at zero temperature. 

In Eqs (2.169) and (2.171) p = —ihd/dx = —ihd/dx. The Hamiltonian (2.169) is thus shown to represent a harmonic oscillator in the absence of external field with an energy spectrum that is shifted uniformly by the last term on the right of (2.171), and whose equilibrium position is shifted according to Eq. (2.170). The new eigenstates are therefore shifted harmonic oscillator wavefimctions ... 

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