Conservation commutator

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 double commutator [[g, Tr /) (/], Tlp q may form new operators different from Q, and some of these new operators may not even be physical observables. When the double commutator conserves the operator Q, one speaks of the auto-correlation mechanism. Otherwise, one speaks of the cross-relaxation process. In other words, cross-relaxation is independent of the nature of the relaxation mechanism, but involves the interconversion between different operators. To facilitate such a possibility, it is desirable to write the density operator in terms of a complete set of orthogonal basis... 

One of the pedagogically unfortunate aspects of quantum mechanics is the complexity that arises in the interaction of electron spin with the Pauli exclusion principle as soon as there are more than two electrons. In general, since the ESE does not even contain any spin operators, the total spin operator must commute with it, and, thus, the total spin of a system of any size is conserved at this level of approximation. The corresponding solution to the ESE must reflect this. In addition, the total electronic wave function must also be antisymmetric in the interchange of any pair of space-spin coordinates, and the interaction of these two requirements has a subtle influence on the energies that has no counterpart in classical systems. 

The theoretical basis for a conserved quantity is the commutation of an effective Hamiltonian with the elements of some symmetry group. If this condition exists, then the irreducible representations of the group are good quantum numbers, i.e., are conserved. Conversely, the existence of good quantum numbers implies a Hamiltonian which commutes with an appropriate group. The most general molecular A-electron Hamiltonians... 

Those systems for which spin is conserved are those systems which are well described by a spin-free Hamiltonian. The spin-free Hamiltonian commutes with the symmetric group 5 F of permutations on electronic spatial indices. It follows that irreducible representations of this symmetric group are good quantum numbers. Certain irreducible representations of S F will be found to correspond to spin quantum numbers. 

Kadanoff and Swift have considered that the time evolution of the state is described by the Liouville equation. They also wrote down conservation equations for the number, momentum, and energy density similar to the ones given by Eqs. (1)—(3). The only difference was that in the treatment of Kadanoff and Swift the densities and currents are operators. The time derivative of the densities are replaced by the commutator of the respective density operator and the Liouville operator, L (as the Liouville operator governs the time evolution). The suffix op in the following equations stands for operator ... 

By analogy with non-exchanging spin systems the superoperators which commute with both the super-Hamiltonian and the superoperator T in composite Liouville space may be called the constants of the motion. In some instances there may be additional constants of the motion which result from the conservation of some molecular symmetry in the exchange, from the magnetic equivalence of some nuclei, and from weak spin-spin coupling. (15, 52) For example,... 

Proposition 4.5 1 Le foncteur j s C — C est pleinement fidfele. commute aiux limites projectives quelconques. II est en particulier exact k gauche et conserve done les structures algebriques dEfinies par limites projectiyes finies. ... 

On remarquera aussi qua le sous-preschema ouvert de G qui a G° pour espace sous-jacent est un sous-groupe de G (que nous notons encore G°) t en effet le morphisme inversion de G conserve 1 image de P et commute les composantes connexes de G qui rencontrent cette image il suffit done de montrer que 1 s G G — G applique G° G° dans G° et pour cela on peut supposer que A est un corps algdbriquement clos (avec les notations de 4.1, G°s identifie en effet au sature de (G k)° par la relation d equivalence definie par 1 homomorphisme u lE) si G8 et G° sont alors des composantes connexes de G , G x G0 est connexe et son image par rencontre 1 image de P par consequent,... 

The norm of the density operator o- = (Tr o-V ) is always conserved under the unitary transformation of Eq. (71). However, in general, additional constants of motion exist. If the effective mixing Hamiltonian is composed exclusively of zero-quantum operators, it commutes with the z component of the total angular momentum operator ... 

Conserved operators commute with the Hamiltonian. We want to know if the orbital angular momentum L (3.59) is conserved. Consider Lx, using the commutation relations (3.6) and equn. (3.161). 

As easily checked, this operator commutes with Hj, leading to the local conservation law of pseudospin angular momentum in the electron-phonon coupled system. If (4) is assumed, the total pseudospin rotation operator T = j) conserved in... 

The Hamiltonian H is a trivial nullifier of Dirac s commutator in this approach, so H is a conserved observable of motion, as was requested. To summarize, we get Heisenberg s... 

Conservation of Commutation Relations of Special Operators and Applications... 

Previous work has not investigated if commutation relations are conserved upon transformation to effective operators. Many important consequences emerge from particular commutation relations, for example, the equivalence between the dipole length and dipole velocity forms for transition moments follows from the commutation relation between the position and Hamiltonian operators. Hence, it is of interest to determine if these consequences also apply to effective operators. In particular, commutation relations involving constants of the motion are of central importance since these operators are associated with fundamental symmetries of the system. Effective operator definitions are especially useful... 

Some semi-empirical theories of chemical bonding assume without proof that the commutation relations between particular model operators are the same as the ones between the corresponding original operators [60-62]. Criteria for the validity of this assumption are investigated here for the first time based on our results for conservation of commutation relations by effective operators. 

The effective operator A is the state-independent part of the definition AL/3, i = I-III. The operator A can thus be obtained by combining the perturbation expansions of its normalization factors and of A into a single expression [73] or by computing these normalization factors and A separately. These combined and noncombined forms of A[, may differ when computed approximately (see Section VI and paper II). The calculation of with the noncombined form is the same as with A since the model eigenvectors used with A are obtained by multiplying those utilized with A[,p by the above normalization factors. The operators and A are nevertheless different and, thus, do not have necessarily the same properties, for example, the conservation of commutation relations studied in Section IV. 

This section studies the previously unaddressed problem of commutation relation conservation upon transformation to effective operators [77]. State-independent effective operators are treated first. 

Section II and Table I show that state-independent effective operators can be obtained with norm-preserving mappings K, or with any of the three kinds of non-norm-preserving mappings K, L), K, L), and K, L). This section first proves that the commutation relations between two arbitrary operators cannot generally be conserved upon transformation to any of these state-independent effective operators. A determination is then made of operators whose commutation relations are preserved by at least some state-independent effective operator definitions, and a few applications are then presented. Particular interest is focused on operators which commute with H, including constants of the motion. 

Theorem IV pertains to commutation relations for all possible state-independent effective operators Let A and B be two arbitrary operators and let F be their commutator F = [A,B. The commutation relation between A and B is, in general, not conserved upon transformation to state-independent effective operators. This theorem is first demonstrated... 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

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