Hamiltonian diagonalization

Then, the canonical transformation makes the driven Hamiltonian diagonal, leading to... 

For longer interaction times (gt >1), there are no exact analytical solutions, thus the numerical analysis has to be applied. We have used two methods to study the quantum dynamics (1) the well-known Hamiltonian diagonalization proposed by Walls and Barakat [24] and (2) the method of global characteristics based on manipulation with spectra [20]. These methods can be applied for arbitrary initial photon statistics. Nevertheless, for the purpose of our chapter, we restrict our analysis to the initial coherent fields solely. Owing to computa-... 

Hamiltonian //. Diagonalization of H gives the exact quantum-mechanical energy levels if the complete set of if - is used, and approximate solutions if the basis is truncated. Instead of focusing on the exact eigenvalues of H, however, the objective of perturbation theory is to approximate them. The starting point is the matrix representation of Hq and V, which will be designated as h... 

It is appropriate to refer these quantities to the molecular-principal-axis system (o, b, c). In this way, the tensor components Dj., g -, and uj,. become uniquely defined molecular properties (g, g = , b, c). However, only the diagonal tensor components are useful in analysing hfs spectra as non-diagonal terms, g+g, do not enter matrix elements diagonal in J, and higher-than-first-order rotational perturbations of JT are barely observable. Thus it is sufficient to consider an effective Hamiltonian diagonal in J. This Hamiltonian is customarily written in terms of rotationally dependent interaction strengths g(A-, ... 

To obtain a simple description of one-electron atoms we must find wavefunctions which are simultaneous eigenfunctions of the main Hamiltonian equation (3.9), and of the much smaller spin-orbit interaction,, equation (3.59). An alternative way of stating this problem is that we are seeking a representation (i.e. a set of wavefunctions) in which the total Hamiltonian diagonal. 

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