Invariance translation

Let us introduce the coherent potential Vk(E) which is thought to be dependent on energy E and exciton momentum k. The coherent potential is translational invariant in the site representation. The Hamiltonian (1) is transformed with the coherent potential taken into account as... 

It is an extremely difficult task to establish generally valid sufficient conditions for roughness-induced wetting. This is a direct consequnce of the loss of translational invariance in such systems. A vast majority of the hitherto performed calculations [185,201] have been based on a simplified model of a rough substrate which assumes periodic variation of the substrate surface location. 

To calculate the profiles and the differential capacitance of the interface numerically we have to choose a differential equation solver. However, the usual packages require that the problem is posed on a finite interval rather than on a semi-infinite interval as in our problem. In principle, we can transform the semi-infinite interval into a finite one, but the price to pay is a loss of translational invariance of the equations and the point mapped from that at infinity is singular, which may pose a problem on the solver. Most of the solvers are designed for initial-value problems while in our case we deal with a boundary-value problem. To circumvent these inconveniences we follow a procedure strongly influenced by the Lie group description. 

This method has already been used successfully for metallic iron. It presents several important advantages. It is a real space calculation for which no translational invariance is required. Calculation up to 500 eV above the edge can be performed because the basis set increases proportionally with the photoelectron energy. The separation between the three contributions (ai, Ojg and ajn) gives also new insight into the physics that is at the origin of XMCD. 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

The first equation gives the diserete version of Newton s equation the second equation gives energy c onservation. We make two comments (1) Notice that while energy eouseivation is a natural consequence of Newton s equation in continuum mechanics, it becomes an independent property of the system in Lee s discrete mechanics (2) If time is treated as a conventional parameter and not as a dynamical variable, the discretized system is not tiine-translationally invariant and energy is not conserved. Making both and t , dynamical variables is therefore one way to sidestep this problem. 

Then the x integration can readily be carried out since by virtue of translation invariance... 

To lowest order in the external field, A%, the scattering is thus determined by the matrix element of the current operator ju(x) between the initial and final one-particle states, p, > and pV>. Let us consider this matrix element in greater detail. Translation invariance asserts... 

However, using translation invariance and the representation (11-655), the left-hand side of this last expression is also equal to... 

If a random process is described as statistically homogeneous, then its statistical moments are translation invariant, i.e., they do not depend on the positions X and X2 only on their difference. Thus the correlation reduces to ... 

Thus the mutual intensity at the observer is the Fourier transform of the source. This is a special case of the van Cittert-Zernike theorem. The mutual intensity is translation invariant or homogeneous, i.e., it depends only on the separation of Pi and P2. The intensity at the observer is simply / = J. Measuring the mutual intensity will give Fourier components of the object. 

Unfortunately, the requirements for translational invariance of the wavelet decomposition are difficult to satisfy. Consequently, for either discretization scheme, comparison of the wavelet coefficients for two signals may mislead us into thinking that the two trends are different, when in fact one is simply a translation of the other. 

Ruanaidh J. Pun T. (1998). Rotation, scale and translation invariant spread s pectrum digital image watermarking. Signal Process, 66(3), 303-317. 

The dipole moment of a charged system is not translationally invariant, and it must be evaluated with the origin at the center of the electric charge, in order to be consistent with the spherical cavity assumption. The dipole moment is therefore computed according to ... 

The exact 1-electron Hamiltonian for a DBA can be written as the sum of the Hamiltonian for a translationally invariant solid plus that for the random perturbations, i.e.,... 

The first summation incorporates the (non-random) translational invariance, while the second includes random deviations from the lattice on a site-by-site basis. Note that the second summation explicitly indicates that all randomness or disorder is diagonal, not off-diagonal. The corresponding exact GF G = (lu — Hyl satisfies the matrix equation... 

To this point, the formalism has been quite general, and from here we could proceed to derive any one of several single-site approximations (such as the ATA, for example). However, we wish to focus on the desired approach, the CPA. To do so, we recall that our aim is to produce a (translationally invariant) effective Hamiltonian He, which reflects the properties of the exact Hamiltonian H (6.2) as closely as possible. With that in mind, we notice that the closer the choice of unperturbed Hamiltonian Ho (6.4) is to He, then the smaller are the effects of the perturbation term in (6.7), and hence in (6.10). Clearly, then, the optimal choice for H0 is He. Thus, we have... 

Due to the translation invariance of the forces, the Fourier matrix elements of the interactions dL and LE obey very simple selection rules. 

In deriving these results we had to remember the remark following (A.46) and to take into account the translational invariance of intermolecular forces. This explains why all the Fourier intermediate states in (A.48) and (A.49) have zero wave vectors. 

It may come as a surprise to some that two commensurate surfaces withstand finite shear forces even if they are separated by a fluid.31 But one has to keep in mind that breaking translational invariance automatically induces a potential of mean force T. From the symmetry breaking, commensurate walls can be pinned even by an ideal gas embedded between them.32 The reason is that T scales linearly with the area of contact. In the thermodynamic limit, the energy barrier for the slider to move by one lattice constant becomes infinitely high so that the motion cannot be thermally activated, and hence, static friction becomes finite. No such argument applies when the surfaces do not share a common period. 

When integrating the equations of motion, it is important to not impose the shear only at the boundaries because this would break translational invariance. Instead, we need to correct the position in the shear direction at each MD step of size At. This correction is done, for instance, in the following fashion ... 

It should be obvious that if A vanishes, the phase degree of freedom has to become redundant, as seen later. It would be worth mentioning that similar configuration has been studied in other contexts [21-23], Note that the configuration in (44) breaks rotational invariance as well as translational invariance, but the latter invariance is recovered by an isospin rotation [26]. 

This section deals with the dynamics of collective surface vibrational excitations, i.e. with surface phonons. A surface phonon is defined as a localized vibrational excitation of a semi-infinite crystal, with an amplitude which has wavelike characteristics parallel to the surface and decays exponentially into the bulk, perpendicular to the surface. This behavior is directly linked to the broken translational invariance at a surface, the translational symmetry being confined here to the directions parallel to the surface. 

Solid surfaces of single crystals provide to some extent the realization of a well-defined two-dimensional (2D) periodic array of atoms. However, the loss of vertical translational invariance at the surface changes the local force held with respect to the bulk forces. As seen in section 3 the charge redistribution is... 

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