Electromagnetism time-dependent expressions

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

The next and necessary step is to account for the interactions between the quantum subsystem and the classical subsystem. This is achieved by the utilization of a classical expression of the interactions between charges and/or induced charges and a van der Waals term [45-61] and we are able to represent the coupling to the quantum mechanical Hamiltonian by interaction operators. These interaction operators enable us to include effectively these operators in the quantum mechanical equations for calculating the MCSCF electronic wavefunction along with the response of the MCSCF wavefunction to externally applied time-dependent electromagnetic fields when the molecule is exposed to a structured environment [14,45-56,58-60,62,67,69-74],... 

We will in this section consider the mathematical structure for computational procedures when calculating molecular properties of a quantum mechanical subsystem coupled to a classical subsystem. Molecular properties of the quantum subsystem are obtained when considering the interactions between the externally applied time-dependent electromagnetic field and the molecular subsystem in contact with a structured environment such as an aerosol particle. Therefore, we need to study the time evolution of the expectation value of an operator A and we express that as... 

The sum-over-states expressions that we have presented in Eqs. (69), (70), and (71) are only true for exact wave functions and they are rather cumbersome methods for calculating time-dependent electromagnetic properties of a quantum mechanical subsystem within a structured environment. The advantage of the sum-overstates expressions is that they illustrate the type of information that is obtainable from response functions. We have utilized modem versions of response theory where the summation over states is eliminated when performing actual calculations, that involve approximative wave functions [21,24,45-47,80-83]. 

In order to obtain a theoretical expression for the oscillator strength, perturbation theory may be used to treat the interaction between electromagnetic radiation and the molecule. Since an oscillating field is a perturbation that varies in time, time-dependent perturbation theory has to be used. Thus, the Hamiltonian of the perturbed system is = 0) +... 

The right-hand side term is the sum of the energies of the electric and magnetic fields. It expresses the time dependence of the energy of the electromagnetic wave, hence, the parameter S represents the energy flow per unit time and area. 

Theoretical expressions of hyperpolarizability coefficients can be obtained by quantum mechanical methods (time dependent perturbation theory). In fact, the oscillating electric field of the incident electromagnetic wave, Eq cos cot, (Eq is the amplitude and co the frequency) can be considered as a time dependent perturbation acting on the molecule. Using second order perturbation theory, the following expression is obtained for Pyk (SHG effect is considered) [6]... 

In the last three sections we have considered the effect of a time-dependent external electric field r,t) and a magnetic induction B r,t) on the motion of an electron and denoted the corresponding potentials with 4> r,t) and A r,t). In the present section we want to collect all the terms and derive our final expression for the molecular electronic Hamiltonian. However, we will not restrict ourselves to the case of external fields because in the following chapters we want to study also interactions with other sources of electromagnetic fields such as magnetic dipole moments and electric quadrupole moments of the nuclei, the rotation of the molecule as well as interactions with field gradients. Therefore, we do not include the superscripts B and on the vector and scalar potential in this section. On the other hand, we will assume that the perturbations are time independent. The time-dependent case is considered in Section 3.9. 

In order to derive a quantum mechanical expression for the frequency-dependent polarizability we can make use of time-dependent response theory as described in Section 3.11. We need therefore to evaluate the time-dependent expectation value of the electric dipole operator (4 o(i (f)) Pa o( (t))) in the presence of a time-dependent electric field, Eq. (7.11). Employing the length gauge, Eqs. (2.122) - (2.124), which implies that the time-dependent electric field enters the Hamiltonian via the scalar potential in Eq. (2.105), the perturbation Hamilton operator for the periodic and spatially uniform electric field of the electromagnetic wave is given as... 

In the beginning of this part it was discussed that one way to obtain practical expressions for the electromagnetic properties, which can be implemented in computer programs, is to make approximations to the exact expressions derived with time-independent and time-dependent perturbation theory in Chapter 3. 

Consider two distinct electromagnetic situations within a fiber. In the first situation, an arbitrary current distribution J gives rise to electric and magnetic fields Eand H, and in the second situation, a point current dipole J results in fields E and H. All six electromagnetic quantities include the implicit time dependence exp (—tot). The arbitrary location and orientation of the dipole is expressed by... 

Consider, for instance, the time-periodic, spatially dependent vector-potential of an incident electromagnetic field, propagating along the z-axis. The Floquet expression for the vector potential of the nth harmonic emitted in the incident field propagation direction (z) is given by... 

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