Effective Hamiltonian time-independent operator

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

Effective Hamiltonians and Effective Operator Definitions Corresponding to a Time-Independent Operator A... 

This demonstrates again the special form of the propagator, a doubly periodic operator times an unitary operator defined by a time independent effective Hamiltonian. 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

As we have seen previously, closed shell systems with an even number of electrons are invariant under the inversion of the time-coordinate. In the time-independent theory this could be related to the anti-unitary operator, K, that commutes with the DCB Hamiltonian. Its effect on the Tj operator is given by... 

On the other hand, approximations to Eq. (8) and time-integration techniques, suitable especially for time-independent Hamiltonians, under the requirement of only a few degrees of freedom and short-time evolution, have been developed and applied extensively in connection with grid-type techniques (see Section 2), by focusing on appropriate algebraic expansions of fhe exponenfial form. For example, such a approach is effected by the split-operator method [4] and references there in. 

Solution of the Lippmann-Schwinger-like equation in Brillouin-Wigner form, equation (55), for the reaction operator followed by solution of the eigenvalue problem (49) for the effective hamiltonian given in equation (52) is entirely equivalent to the solution of the time-independent Schrodinger equation, equation (1), for the state a. Furthermore, although recursion leads to the expansion (56), equation (55) remains valid when the series expansion does not converge. Equation (55) can be written... 

The Hamiltonian (4.1) is approximate as it does not take into account the spin-orbit interaction and other relativistic effects. The calculation of eigenfunctions and eigenvalues of the operator (4.1), i.e. the solution of the time-independent Schrodinger equation... 

The time-independent Hamiltonian JCq is itself the sum of an operator which determines the unperturbed eigenvalues and eigenfunctions of the atomic electrons, and an operator 3fpert> hich describes the interaction of the atom with static external magnetic fields. In Chapters 16 and 17 additional terms will be introduced into equation (15,16) to account for the effects of magnetic resonance and optical pumping respectively. 

A chemical molecule, by contrast consists of many particles. In the most general case N independent constituent electrons and nuclei generate a molecular Hamiltonian as the sum over N kinetic energy operators. The common wave function encodes all information pertaining to the system. In order to constitute a molecule in any but a formal sense it is necessary for the set of particles to stay confined to a common region of space-time. The effect is the same as on the single confined particle. Their behaviour becomes more structured and interactions between individual particles occur. Each interaction generates a Coulombic term in the molecular Hamiltonian. The effect of these terms are the same as of potential barriers and wells that modify the boundary conditions. The wave function stays the same, only some specific solutions become disallowed by the boundary conditions imposed by the environment. 

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