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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.
The Bloch states ψ k (r) are defined as the eigenfunctions of a particular Hamiltonian, and are therefore defined only up to an overall phase. By applying a phase transformation e iθ ( k ) to the functions ψ k ( r ), for any (real) function θ ( k ), one arrives at an equally valid choice.
Bloch sphere. In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch. [1] Mathematically each quantum mechanical system is associated with a separable complex Hilbert space.
The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.
The solutions in this case are known as Bloch states. Since Bloch's theorem applies only to periodic potentials, and since unceasing random movements of atoms in a crystal disrupt periodicity, this use of Bloch's theorem is only an approximation, but it has proven to be a tremendously valuable approximation, without which most solid-state ...
The periodicity of the crystalline potential allows the application of the Bloch theorem, which states that the Hamiltonian eigenstates take the form = (), where is a band index, is a wavevector in the reciprocal-space (Brillouin zone), and () is a periodic function of .
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: