Pauli, generally matrices

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

Of course, the Coulomb interaction appears in the Hamiltonian operator, H, and is often invoked for interpreting the chemical bond. However, the wave function, l7, must be antisymmetric, i.e., must satisfy the Pauli exclusion principle, and it is the only fact which explains the Lewis model of an electron pair. It is known that all the information is contained in the square of the wave function, 1I7 2, but it is in general much complicated to be analyzed as such because it depends on too many variables. However, there have been some attempts [3]. Lennard-Jones [4] proposed to look at a quantity which should keep the chemical significance and nevertheless reduce the dimensionality. This simpler quantity is the reduced second-order density matrix... 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

However, as distinguished from (29), it cannot be transformed to the simple diagonal form. With noncommutative Pauli matrices and ax in the exponent, the operator S in (30) cannot be presented in analytic form as a diagonal matrix with shift operators in its main diagonal, though it can be brought to a closed general 2x2 matrix form, whose matrix elements contain the above derivatives. 

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