Bloch eigenstate

A recent trend has been to use Wannier functions to calculate properties. Wannier functions are orthonormal localized functions spanning the same space as the eigenstates of a particular band or group of bands and are Fourier transforms of the Bloch eigenstates. For one band, i, Wannier functions, w, are given by... 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

The basic element of a quantum computer is the quantum bit or qubit. It is the QC counterpart of the Boolean bit, a classical physical system with two well-defined states. A material realization of a qubit is a quantum two-level system, with energy eigenstates, 0) and 1), and an energy gap AE, which can be in any arbitrary superposition cp) = cos(d/2) 0) + exp(i0)sin(0/2) l).These pure superposition states can be visualized by using a Bloch sphere representation (see Figure 7.1). 

The simplest optical Bloch equations result from a system comprised of two eigenstates j), E2) of the molecule Hamiltonian HM that experience the electric field-dipole interaction... 

In contrast, in CsCl wc sufficiently simplified the matrix elements that we were able to use single Bloch sums, of the kind in Eq. (3-19), as eigenstates we cannot do that here. 

FIGURE 5 A thermal density operator can be decomposed into pure states in infinitely many different ways. Mbdng the vectors corresponding to eigenstates or chiral states or alternative chiral states with 50% probability always leads to the zero vector, i.e., the center of the Bloch sphere, corresponding to the density operator Dp =... 

In perfect crystals, Bloch s theorem applies for the calculation of electronic states as well as for elementary excitations such as phonons or magnons. Electronic states, for example, can be described as eigenstates of the system with a well-... 

In disordered metals, the electrons couple with the static structure at any. K-value and are heavily damped close to pseudo Brillouin-zone boundaries. A definite dispersion relation does not exist (Fig. 5.3bl) and Bloch s theorem is no longer valid. Electronic states cannot be described as eigenstates of the system and, strictly speaking, cannot be translated to the reduced-zone scheme. 

In the BCS theory, the normal (non-superconducting) state of the metal is described by Bloch single particle eigenstates (8) k> labelled by a wavevector k,... 

Figure 25.10 Calculated eigenvalues and eigenstates lead to a series of vibrational states with quantum numbers (n, k) for the k = 0 state the Bloch orbitals of each branch were examined. At low energies, the respective orbitals are mainly confined around the fee, top, or hep sites of the Pt(l 11) surface. The figure shows for selected bands the localization/delocalization ofthe orbitals n = 1 (Fig. 25.9(a, b)) n = 3 (Fig. 25.9 (c, d)) n = 4 (Fig. 25.9 (e, f)) n = 26 (Fig. 25.9 (g, h)) n = 15 (Fig. 25.9 (k, I)), and n = 16 (Fig. 25.9 (m, n)), whereby the right sequence of graphs displays the probability density p(r) in transversal sections. Ten equidistant contour lines are used in each graph. After Badescu et al. [55].

Sg = trg(ff/Og0 is the QuANTON S initial Bloch vector. The z-axis is chosen for quantization the eigenstates z) of this z-basis are pure states corresponding to... 

Note that 7i r, t) is periodic both in time and space hence, it is referred to as a Roquet-Bloch (TO) Hamiltonian. In other words, the Roquet states can be expressed as FB eigenstates. 

An appropriate and useful approach to follow the evolution of a quantum state is the Bloch sphere representation, introduced in Chapter 3. This is a geometrical scheme in which the quantum state and its evolution is represented by the trajectory of a vector over the so-called Bloch sphere (Figure 4.13). In the Bloch sphere, the poles represent the two eigenstates of the system, whereas the equatorial plane corresponds to an uniform superposition of these two eigenstates. 

The investigation of the localization properties (Anderson localiza-tion ) of the eigenstates of disordered chains is important primarily from the standpoint of their transport properties. If the Fermi eneigy (ep) falls into a more or less continuous region of allowed energy levels, one has to know whether the states around p are localized or delocalized. If the wave functions are delocalized, a coherent, Bloch-type conduction is still possible. If, however, they are localized, one can expect only incoherent, hopping-type chaise transport. 

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