Hamiltonian Transition Expectation Values

Physically, we consider the parametric regime where coupling is so weak that it hardly induces any transition during At and, consequently, the effective Hamiltonian has a nondemolition form in the sense defined in the previous section. Since the interaction term commutes with both system Hamiltonians, the (expectation values of the) subsystem energies are constants of time. This is, in fact, a consequence of the assumed lack of resonances between the oscillator and the spin. 

While the two methods are, at face value, quite different in the ways in which full quantum dynamics is reduced to quantum-classical dynamics, there are common elements in the manner in which they are simulated. The Trotter-based scheme for QCL dynamics makes use of the adiabatic basis and is based on surface-hopping trajectories where transitions are sampled by a Monte Carlo scheme that requires reweighting. Similarly, ILDM calculations make use of the mapping hamiltonian basis and also involve a similar Monte Carlo sampling with reweighting of trajectories in the ensemble used to obtain the expectation values of quantum operators. 

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. 

The transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wavefunction. As discussed in detail in section 2, the four-component Dirac spinor will have only two nonvanishing components, as soon as the complete decoupling of the electronic and positronic degrees of freedom is achieved, and can thus be used as a two-component spinor. This feature can be exploited to calculate expectation values of operators in an efficient manner. However, this procedure requires that some precautions need to be taken care of with respect to the representation of the operators, i.e., their transition from the original (4 x 4)-matrix representation (often referred to as the Dirac picture) to a suitable two-component Pauli repre-... 

The second-order energy correction thus consists of two terms a ground-state expectation value over the second-order Hamiltonian and a so-called sum-over-states term, which involves a summation over all excited states of the system and transition moments between the ground state and these excited states with the first-order Hamiltonian Finally, we can insert the expressions for the first- and second-order perturbation Hamiltonians, Eq. (2.108),... 

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