Search results
Results from the WOW.Com Content Network
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,
The eigenstates of the finite system can be found in terms of the Bloch states of an infinite system with the same periodic potential. If there is a band gap between two consecutive energy bands of the infinite system, there is a sharp distinction between two types of states in the finite lattice.
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'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:
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.
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 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 theorem arises as a direct consequence of the aforementioned fact that the lattice symmetry translation operator commutes with the system's Hamiltonian. [ 3 ] : 261–266 [ 5 ] One of the notable aspects of Bloch's theorem is that it shows directly that steady state solutions may be identified with a wave vector k {\displaystyle \mathbf {k ...