![]() They avoid each other by differing their energy eigenvalues. The reason why electrons do not collapse to one flat band because electrons obey the Pauli principle. Remember the energy is also a quantum number. Two Fermions cannot have all quantum numbers with the same values. The Pauli principle is indeed very important in explaining how the band structure forms the way it does. And because they are indistinguishable the Pauli Exclusion Principle causes half of all the valence electrons to be in different energy states and form a band. ![]() The same holds for all other valence electrons. For example, the valence electron initially in $A$ can (when brought together with the other atoms to form a chain) be found in $C$, $D$.etc. You can't tell anymore where every electron in this chain is to be found. ![]() The atom wavefunctions do not overlap, but the wavefunctions of all valence electrons form one "overarching" wavefunction. I think here lies the core of the problem. If yes, can we show that it is equivalent to the well-known periodical-Hamiltonian demonstration? Is this Pauli principle approach really correct? If yes, what did I misunderstand? I find it highly surprising to find an explanation based on a radically different principle, and I have a hard time imagining that both ideas are equivalent. I understand that the exclusion principle will be important to describe how those bands get filled, but it should not be necessary, in my understanding, to explain that they exist. ![]() It only relies on the translational symmetry of the Crystal, deducing the Bloch states and injecting them in the SE to show the bands $E_n(k)$ emerging as a solution. Actually, we could establish the band structure in a single-electron approximation, which suggests that the exclusion principle would have Nothing to do with this result. The way to introduce band structure that I have studied does not make use of the Pauli Principle (in fact, in the book I am thinking of, a chapter on the band structure is placed before the one tackling identical particles). nothing to do with each other as the distance increases? (ignoring the fact that they could eventually be delocalized as conduction electrons in a metallic crystal) Why would the Pauli principle cause a never-ending splitting as we add more and more atoms to the crystal whose electrons have. No need to call on the Pauli principle, which in this case, should provide no information. Being "distant from each other", the spatial part of $\psi$ is distinct for those electrons, and therefore the states are already different. However, why would the Pauli principle say anything about the electrons of A and C? I mean, the wavefunctions of the electrons of A and C do not overlap. I could more or less imagine that the overlap of the valence orbitals of A and B could imply that the Pauli exclusion comes into play for the valence electrons in the A-B system (although I am already not sure this is the right way to describe this). I mean, to illustrate, if we imagine a 1D monoatomic crystal and take a string of 3 atoms that we label A, B, and C: Therefore, I don't see how the Pauli principle could Apply to the orbitals of two atoms "far away" in the Crystal. The state of a particle is described by its wave function. For me, the Pauli exclusion principle states the Following:Ģ identical fermions in the same physical system can not be in the same state at the same time The energy of adjacent levels is so close together that they can be considered as a continuum, an energy band. Since the number of atoms in a macroscopic piece of solid is a very large number (N~1022) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10−22 eV). Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, each with a different energy. If a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap. They try to give an intuitive explanation of the band structure relying heavily on Pauli Exclusion Principle: two fermions cannot be in the same quantum state: the Pauli exclusion principle.Checking the wikipedia article on band structure, I got caught in major doubts. Thus there is no possible antisymmetric combination involving identical states, i.e. \) correspond to different configurations and are usually set to zero.
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