Bloch theorem Schrodinger equation

For a periodic lattice, it can be shown (Bloch theorem) that the solutions to the one-electron Schrodinger equation are of the... 

Our lengthy discussion of k space began with Bloch s theorem, which tells us that solutions of the Schrodinger equation for a supercell have the form... 

The Bloch theorem is one of the tools that helps us to mathematically deal with solids [5,6], The mathematical condition behind the Bloch theorem is the fact that the equations which governs the excitations of the crystalline structure such as lattice vibrations, electron states and spin waves are periodic. Then, to jsolve the Schrodinger equation for a crystalline solid where the potential is periodic, [V(r + R) = V(r), this theorem is applied [5,6],... 

Have the correct form to be a solution of Equation 1.8. As a result, the Bloch theorem affirms that the solution to the Schrodinger equation may be a plane wave multiplied by a periodic function, that is [5,6],... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

The wave function has the same amplitude at equivalent positions in each unit cell. Thus, the full electronic structure problem is reduced to a consideration of just the number of electrons in the unit cell (or half that number if the electronic orbitals are assumed to be doubly occupied) and applying boundary conditions to the cell as dictated by Bloch s theorem (Eq. 4.14). Each unit cell face has a partner face that is found by translating the face over a lattice vector R. The solutions to the Schrodinger equation on both faces are equal up to the phase factor exp(zfe R), determining the solutions inside the cell completely. 

This is the parabolic curve shown in Fig. 4.4(a). A linear atomic lattice will provide a periodic rather than a constant potential, i.e. V(x) = V[x + a), where a is the repeat distance of the array. Bloch s theorem, also known as Floquet s theorem, states that possible solutions of the Schrodinger equation with a periodic potential are ... 

The presence of the periodic potential f/(r) has important consequences with regard to the solutions of the time-independent Schrodinger equation associated with the Hamiltonian (4.71). In particular, a fundamental property of eigenfunctions of such a Hamiltonian is expressed by the Bloch theorem. 

The band structure of crystalline solids is usually obtained by solving the Schrodinger equation of an approximate one-electron problem. In the case of non-metallic materials, such as semiconductors and insulators, there are essentially no free electrons. This problem is taken care of by the Bloch theorem. This important theorem assumes a potential energy profile V(r) being periodic with the periodicity of the lattice. In this case the Schrodinger equation is given by... 

It can be shown that the choice of wave vector q is not unique, but that every wave vector q = q + b, where b is any reciprocal wave vector, also satisfies the Bloch theorem. You usually choose the wave vector which lies in the Brillouin zone and represent the state by that wave vector. This now means that we can solve the Schrodinger equation just for the Brillouin zone and do not have to solve for the whole lattice. 

For each k value, there are n solutions of the Schrodinger equation. Thus, using Bloch s theorem... 

When it comes to the course and dispersion of more complicated bands, this is easily illustrated by two other one-dimensional examples. Note that the above Bloch formula for the construction of tp k) at some k value did not depend on the orbital involved the plus/minus sign changes only resulted from the exponential pre-factor. Since Bloch s theorem just depends on some solution of the Schrodinger equation, and this may be another atomic orbital or, equally well, a molecular orbital, let us first assume, in Scheme 2.2, a onedimensional chain of, say, nitrogen atoms where each N carries a set of one 2s... 

In (6.5) the subscript n indicates the band index and fe is a continuous wave vector that is confined to the first Brillouin zone of the reciprocal lattice. The index n appears in the Bloch theorem because for a given k there are many solutions to the Schrodinger equation. Because the eigenvalue problem is set in a fixed finite volume, we generally expect to find an infinite family of solutions with discretely-spaced eigenvalues which we label with the band index n. The wave vector k can always be confined to the first Brillouin zone. The vector k takes on values within the Brillouin zone corresponding to the crystal lattice, and particular directions like r,A,A,Z (see Figures 4.13-4.15). 

Figure 4.13. Electronic states in the nearly free electron model for a 1-D chain with unit ceU length, a. The energy states in (a) represent solutions to the Schrodinger equation (via the Bloch theorem) with an infinitesimal potential. In contrast, solution (b) represents a finite lattice potential, which results in the formation of gaps between the parabolas at the BZ boundary. That is, there is no longer a continuum of states from the lowest to infinity, but regions where no allowed states may exist. Reproduced with permission from Hofmann, P. Solid State Physics An Introduction, Wiley New York, 2008. Copyright 2008 Wiley-VCH Verlag GmbH Co.

According to Bloch s theorem, for a periodic potential V z) the solutions of the one-dimensional single-particle Schrodinger equation... 

In modem language, Bloch s theorem can be expressed as follows. For a given wave function tp k, r) which fulfills Schrodinger s equation, there exists a vector k such that translation by a lattice vector T is equivalent to multiplication by a phase factor ... 

According to Bloch s theorem [6], the solutions to Schrodinger s equation mnst take the form ... 

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