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$\renewcommand{ket}[1]{|#1\rangle}$ The basic logical connection here is $$\text{symmetry} \rightarrow \text{degeneracy} \rightarrow \text{avoided crossing} \rightarrow \text{band gap} \, .$$ $\textrm{symmetry}\rightarrow \textrm{degeneracy}$ Consider an operator $S$ and let $T(t) = \exp[-i H t / \hbar]$ be the time evolution operator. If $$[ T(t), S] = 0 ... 2 If there is only one band maximum in the BZ, this point is one of the high-symmetry points of the BZ. However, there can be cases where there are many points which are a band maximum and they are not at one of the high-symmetry points of the BZ. These points however are all connected by a symmetry operation. An example of a system with band minima away ... 2 The quantization of energy levels appears both in quantum and classical mechanics, and it is not a consequence of the Schrödinger equation. It is a consequence of confinement. In fact, anytime that a wave equation (any quantum equation for the wavefunction, or a classical equation for a classical field, e.g., EM field) has periodic boundary conditions in ... 2 Quantization is an experimental fact that forced physicists to consider theories that could explain the data. This happened in the beginning of the twentieth century. 1) black body radiation could only be explained by assuming that the radiation came in quanta, i.e. not in a continuous spectrum.) 2) The photoelectric effect showed that light behaved as a ... 0 Consider a Hamiltonian which is translationally invariant. For example, H = \frac{ \hat{p}^2}{2m} = - \frac{1}{2m} \frac{ \partial^2}{\partial x^2}. There are other options (any Hamiltonian which does not contain the operator \hat{x} for example). Such a Hamiltonian forms quantum states which have definite momentum, i.e. they are eigenstates of the ... 0 Bound states have quantized energies, while unbound states have continuous energies. This can be understood by thinking of, for example, the 1D infinite square well. You can think semiclassically of the particle "bouncing back and forth" between the walls of the square well potential. At most wavelengths, the reflected particle will interfere with itself and ... 4 Technically, both solids and gasses have quantized energy levels. The difference is that molecules of a gas interact with other molecules very weakly, so the energy levels observed in emission or absorption of a collection of gas molecules are almost exactly the same as the energy levels that would be observed if you had a single gas molecule in isolation. ... 2 In quantum mechanics the equation of motion is the Schrödinger equation$$ i\hbar\,\frac{\partial}{\partial t}\,|\psi\rangle = H|\psi\rangle  where the (self-adjoint) operator $H$, the Hamiltonian, determines its evolution. The energy levels are, by definition, the eigenvalues of such operator in its domain of definition $\mathcal{D}_H$. Spectral theory ...

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If the energy levels are continuous (within a given interval of energies), then a particle (or system) can in principle have any energy in that interval. If they are quantized into say $E_1,E_2...$, then a particle (or system) can have only one of those energies, and not anything in between them.

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Bands are the permitted answers to the Schrodinger equation concerning a periodic configuration of atoms. The set of forbidden answers also form different bands, which we call band-gaps and they of course fall between the permitted answers. Therefore, technically speaking, we have infinite number of bands as well as an infinite number of band-gaps. Now, when ...

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The bandstructure including conduction and valence band stems from the periodic potential created by the nuclei. Since it depends on the species and particular species have a certain number of electrons, an "empty" electronic bandstructure does not make too much sense. Electrical transport can only take place in partially occupied bands, as electrons or ...

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