Bloch-like equation

The Bloch-like equation in Brillouin-Wigner form... 

In the previous section, we have given the Brillouin-Wigner perturbation expansion for the exact wave function for state a developed with respect to some single reference or multireference model function In this section, we define the Brillouin-Wigner wave operator and the corresponding Bloch-like equation [64]. 

It should be noted that equation (41) is exact. It is entirely equivalent to the Schrodinger equation (1). However, whilst repeated substitution of (41) into itself generates the expansion (40), the Bloch-like equation (41) remains valid when the series expansion (40) is not convergent. Equation (41) can be written in the form... 

Writing the resolvent Ba in the sum-over-states from given in (33), we can write the Bloch-like equation (41) in the form... 

By combining equation (41), the Bloch-like equation in Brillouin-Wigner form, with equation (53), the definition of the reaction operator, a Lipp-mann-Schwinger-like equation [122,123] is obtained... 

Taking the exponential ansatz for the wave operator (60), the Bloch-like equation (41) then becomes... 

The Green s operator satisfies a Bloch-like equation... 

The tails of the localised wavefunctions will overlap, giving rise to a small interaction between neighbouring sites. When x < ajl the difference W(x) = V(x) Uq(x), which is a measure of the inter-site interaction, will be small. A possible solution to the Schrodinger equation is the Bloch-like function ... 

Brillouin- Wigner expansions in quantum chemistry Bloch-like and Lippmanri-Schwinger-like equations... 

BRILLOUIN-WIGNER EXPANSIONS IN QUANTUM CHEMISTRY BLOCH-LIKE AND LIPPMANN-SCHWINGER-LIKE EQUATIONS... 

We do not propose to describe here the details of specific applications of Brillouin-Wigner methods to many-body systems in chemistry and physics. Such details can be found in our article in the Encyclopedia of Computational Chemistry [1] and in our review entitled Brillouin-Wigner expansions in quantum chemistry Bloch-like and Lippmann-Schwinger-like equations [36]. We have established a website at... 

Theoretical level populations. Sinee there are population variations on time seale shorter than some level lifetimes, a complete description of the excitation has been modeled solving optical Bloch equations Beacon model, Bellenger, 2002) at CEA. The model has been compared with a laboratory experiment set up at CEA/Saclay (Eig. 21). The reasonable discrepancy when both beams at 589 and 569 nm are phase modulated is very likely to spectral jitter, which is not modeled velocity classes of Na atoms excited at the intermediate level cannot be excited to the uppermost level because the spectral profile of the 569 nm beam does not match the peaks of that of the 589 nm beam. 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

This section summarizes primarily the classical description of NMR based on the vector model of the Bloch equations. Important concepts like the rotating frame, the effect of rf pulses, and the free precession of transverse magnetization are introduced. More detailed accounts, still on an elementary level, are provided in textbooks [Deri, Farl, Fukl]. 

It can be shown [33] that the eigenfunctions ip (r) of (5.3) are the products of Bloch functions and hydrogen-like envelope functions F (r). It can be further demonstrated that the envelope functions are eigenfunctions of a so-called EM equation 2... 

For the square well, taking into account that a variable separation is possible for a potential of the form V(r) = Vo(0(x)+0(y)), where 0 is the step-like function which is equal to 0 and 1 inside and outside the well respectively, one has two independent ID set of equations. Moreover, assuming that (r) = f(x) f y), the Bloch s boundary conditions can be split into two ones for f x) and as well. Finally, one has two independent ID Kronig-Penney problems for and Ky. Thus, for x-direction ... 

As has already been discussed above, the local basis states Rsu) play a key role in any local orbital scheme. In the RFPLO method they are calculated as the solutions of an atom-like single particle Dirac equation (33) in a spherical, orbital dependent potential (34) and XC-field (35). The atom-like potential U/ contains the spherically averaged crystal potential V/ around the lattice site s, which ensures that a Bloch sum of the core orbitals and the related core eigenvalues are very good approximations to the solutions of the true crystal Hamiltonian. For the calculation of valence states an additional attractive r -potential, acting on the large components only, is applied. 

The discussion makes it obvious that it is a far from trivial problem how to treat r in an actual calculation. Therefore, we decided to first study the Hiickel-like model numerically, using two different approximations for r. In one case we used the full potential of Fig. 1(c), whereas in the other case we considered only the periodic part of Fig. 1(b). As discussed above, the latter case corresponds to use the separation of equations (13) and (15) applied on the Bloch waves formed by the basis functions (i.e., not the eigenfunctions). 

We would like to emphasize that the photon statistics we consider is classical, while the Bloch equation describing dynamics of the SM has quantum mechanical elements in it (i.e., the coherence). In the weak laser intensity case, the Bloch equation approach allows a classical interpretation based on the Lorentz oscillator model as presented in Appendix B.3. 

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). 

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