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Bloch's theorem — For electrons in a perfect crystal, there is a basis of wave functions with the following two properties: each of these wave functions is an energy eigenstate,
Bloch's theorem was inspired by the following theorem of Georges Valiron: Theorem. If f is a non-constant entire function then there exist disks D of arbitrarily large radius and analytic functions φ in D such that f(φ(z)) = z for z in D. Bloch's theorem corresponds to Valiron's theorem via the so-called Bloch's principle.
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice.The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice.
A useful advantage of the Bloch sphere representation is that the evolution of the qubit state is describable by rotations of the Bloch sphere. The most concise explanation for why this is the case is that the Lie algebra for the group of unitary and hermitian matrices S U ( 2 ) {\displaystyle SU(2)} is isomorphic to the Lie algebra of the ...
It is a requirement for both the free electron model and the nearly-free electron model, where it is used alongside Bloch's theorem. [1] In quantum mechanics, this approximation is often used to simplify a quantum many-body problem into single-particle approximations. [1]
Blaschke selection theorem (geometric topology) Bloch's theorem (complex analysis) Blondel's theorem (electric power) Blum's speedup theorem (computational complexity theory) Bôcher's theorem (complex analysis) Bochner's tube theorem (complex analysis) Bogoliubov–Parasyuk theorem (quantum field theory) Bohr–Mollerup theorem (gamma function)
The Bloch's function is an exact eigensolution for the wave function of an electron in a periodic crystal potential corresponding to an energy (), and is spread over the entire crystal volume. Using the Fourier transform analysis, a spatially localized wave function for the m -th energy band can be constructed from multiple Bloch's functions:
Bloch's principle is a philosophical principle in mathematics stated by André Bloch. [1]Bloch states the principle in Latin as: Nihil est in infinito quod non prius fuerit in finito, and explains this as follows: Every proposition in whose statement the actual infinity occurs can be always considered a consequence, almost immediate, of a proposition where it does not occur, a proposition in ...