Hamiltonian matrix fields

A quantitative evaluation of the relaxivities as a function of the magnetic field Bo requires extensive numerical calculations because of the presence of two different axes (the anisotropy and the external field axis), resulting in non-zero off-diagonal elements in the Hamiltonian matrix (15). Furthermore, the anisotropy energy has to be included in the thermal equilibrium density matrix. Figures 7 and 8 show the attenuation of the low field dispersion of the calculated NMRD profile when either the crystal size or the anisotropy field increases. 

Figure 8.1 Schematic structure of the Hamiltonian matrix for a collision system in an external field expressed in the total angular momentum representation.

In the presence of the field, the molecular states are coherent superpositions of the states AM ) A ). In principle, the basis set must include an infinite number of states A ). However, the Floquet Hamiltonian matrix is block-diagonal and the diagonal matrix elements of the Floquet matrix separate in values SiS k-k increases. This suggests that it may be possible to include in the basis set a finite number of states from - max to max seek convergence with respect to In other words, the eigenstates of the Floquet Hamiltonian... 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

Consider a quantum center (i.e., a molecule or a subpart of a molecule) embedded in a classical molecular environment. Defining with rn the nuclear coordinates of the quantum center and with x the coordinates of the atoms providing the (classical) perturbing field we can expand [26] the perturbed Hamiltonian matrix H of the quantum center on the Born-Oppenheimer surface as... 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

In the framework of many-body perturbation theory, one first defines the scattering matrix. S as a time-ordered exponential in terms of the perturbing Hamiltonian and field operators [471. Then, one considers the matrix elements corresponding to the proccs.s in which the recoiless probe particle carries the system either from an initial state a to a final state af >(, (single excitation) or from an initial state to a final state a/a (I>(f... 

It is interesting to consider one particular initial state of the field in which there are only two photons at the fundamental mode and no photons in the second-harmonic mode 2,0). With this initial state the only state that can be created in the process is the state 0,1) with one photon in the second-harmonic mode and zero photons in the fundamental mode. Next, the second-harmonic photon can be downconverted into two photons of the fundamental mode, and we observe fully periodic evolution. The evolution is thus restricted to the two-dimensional subspace 2,0), 0,1). The Hamiltonian matrix in this subspace has the form... 

Fig. III.9. Low J section of the Hamiltonian matrix of an asymmetric top molecule such as for instance ethyleneoxide in the absence of exterior fields. The matrix is set up in the eigenfunction basis of the limiting oblate symmetric top, =...

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