Continuous energy levels

We notice that at low temperatures the specific heat of a system with continuous energy levels, obeying the Fermi statistics, is proportional to the temperature. We shall later see that this formula has applications in the theory of metals. 

In most metals there will be bands derived from the outermost s, p, and d atomic levels, leading to a system of bands, some of which will overlap as described above. Where overlap does not occur, the almost continuous energy levels of the bands are separated by a forbidden zone, or band gap. Only the outermost atomic orbitals form bands the inner orbitals remain localized on the individual atoms and are not involved in bonding. 

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. 

In Fig. 13, the continuous energy levels on the right hand side represent the phonon states of a solid. In a liquid such as NM there are no true phonons, but there is a band of collective excitations of the lower energy continuum that... 

Because of the wide range of very narrowly spaced or continuous energy levels associated with incandescent particles or ion recombinations, only a continuous spectrum of photon energies can be produced. 

For a large time, there is an oscillatory behavior that leaves positive and negative contributions even for small Let us assume a density of states p(E), which is constant in a region around Eg. The rate of decrease of the total probability of the state T g, which was equal to unity at time t = 0, may be obtained by integrating over the range of continuous energy levels (E = ha) ... 

Groups of MOs having elosely spaced (essentially continuous) energy levels (see Fig. 7.27 in the text)... 

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