Translational invariance Hamiltonian

The Hamiltonian /lclcc(f f) has the same invariance under the rotation-reflection group 0(3) as does the full translationally invariant Hamiltonian (6), and it has a somewhat extended invariance under nuclear permutations, since it contains the nuclear masses only in symmetrical sums. Since it contains the translationally invariant nuclear coordinates as multiplicative operators, its domain is of... 

A.2 A permutationally restricted translationally invariant Hamiltonian operator for NH3... 

This particle-hole excitation is illustrated in Fig. 3.4. Now, for translationally invariant Hamiltonians iF is a good quantum number. However, unlike the noninteracting Hamiltonian, the interacting Hamiltonian mixes states with different k. ... 

As we have already discussed in Section 4.11, self-trapping can never be a true consequence of electron-phonon interactions in a translationally invariant Hamiltonian it is an artefact of the adiabatic approximation, which freezes the nuclear degrees of freedom. When the nuclear degrees of freedom are quantized it is possible to construct a translationally invariant wave-packet of both the electron and nuclear degrees of freedom. The band width of this wavepacket is a function of the phonon frequency, and in the adiabatic limit (w —> 0) it will vanish. ... 

Choosing Electronic and Nuclear Variables in the Translationally Invariant Hamiltonian... 

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... 

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 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.,... 

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... 

The excitation stabilizations D and A have been included in haj0 and h i0. In the expression (2.62) we take as energy origin hd>0 + h 0 and we restrict our investigation to states with total wave vector q equal to that of the incident photon, the resulting hamiltonian H remaining translationally invariant. Then it is appropriate to use two new basis sets ... 

We now study the disordered effective hamiltonian (4.4). Since a direct diagonalization of (4.4) is too hard, we shall have to use approximations which are conveniently expressed in the resolvent (or Green s function) formalism. The translation-invariant K sum in HeU is restricted to the optical wave vectors only (for K oj/c, RK / K 0I)- Therefore, it is possible to restrict the problem to this small part of the Brillouin zone using the projector operator... 

With this choice of coordinates, the translationally invariant Coulomb Hamiltonian takes the form ... 

If the f were assigned values, b say, based on choices x" = ag in the laboratory-fixed frame, then this would be the translationally invariant form of the electronic Hamiltonian appropriate to a particular classical nuclear geometry and... 

Whatever way it is intended to go in specifying the electronic coordinates, they must be specifiable as translationally invariant so that the centre of mass motion can be separated from Schrodinger s equation for the system. It is only the translationally invariant part of the Hamiltonian that can have a bound state spectrum and thus be relevant to both the scattering and the bound molecule problem. 

Let us now consider a ferromagnetic system made of spins a (r) attached to the lattice sites. We shall admit that the spin-spin interaction is given by a Hamiltonian JV a) which will be assumed to be translationally invariant. At equilibrium, the weight of a configuration of the system is given by the Boltzmann s factor... 

First consider the coordinate transformation (rj, p ) —> (rj + a, p -) for j — 1,. . . N, where a is an arbitrary vector. This corresponds to a shift or translation in the origin of the coordinate system by the arbitrary vector a. Now in general the intermolecular potential V is a function of the relative positions, (rj — r ) thus under the above transformation Vtfri — tj) -> Vij(T — Tj). This implies that in the absence of external forces, both the Hamiltonian H and its corresponding Liouvillian are invariant to this transformation. We say that H and L are translationally invariant. 

Equation (1.37) cannot be used directly since the degrees of freedom at positions r and r are coupled and the integrals cannot be done. However, for translationally invariant systems, the coupiing only depends on the difference r —r and the Hamiltonian can be diagonalized in Fourier space. One thus considers the transforms ... 

The Non-Relativistic Hamiltonian and Conservation Laws Invariance with Respect to Translation Invariance with Respect to Rotation... 

Two things played important roles in getting this surprising pure-like result (i) The disorder correlation has a statistical translational invariance coming from the delta function in the 2-coordinate, and (ii) the quadratic nature of the Hamiltonian. If disorder had any correlation along the length of the polymer, Eq. (16) will not be valid. 

In all of the following we assume that the Hamiltonian is invariant to these transformations. For example, in the absence of electromagnetic forces the Hamiltonian is a quadratic function of the momentum—hence for time reversal invariance H(, p) = H q —p). In addition, if the potential is only a function of the distances between particles, H is translationally invariant, reflection invariant, and has even parity. Because po(F) po(F) will be invariant to all... 

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