Propagation phase factor

Let us investigate the change of the various components of the density matrix under propagation from time t to t + 5t. We first consider the action of jSfo- Due to this operator, the density matrix element p (t) attains a phase factor... 

It can be shown that the Sagnac effect with platform at rest is the rotation of the plane of linearly polarized light as a result of radiation propagating around a circle in free space. Such an effect cannot exist in the received view where the phase factor in such a round trip is always the same and given by Eq. (554). However, it can be shown as follows that there develops a rotation in the plane of polarization when the phase is defined by Eq. (553). It is now known that the phase must always be defined by Eq. (553). Therefore, proceeding on this inference, we construct plane polarized light as the sum of left and right circularly polarized components ... 

We remark that the simulation scheme for master equation dynamics has a number of attractive features when compared to quantum-classical Liouville dynamics. The solution of the master equation consists of two numerically simple parts. The first is the computation of the memory function which involves adiabatic evolution along mean surfaces. Once the transition rates are known as a function of the subsystem coordinates, the sequential short-time propagation algorithm may be used to evolve the observable or density. Since the dynamics is restricted to single adiabatic surfaces, no phase factors... 

Tlic orbitals entering the A] states propagating in a -direction (upward in the figure), and the phase factors entering for each plane. Oxygen atoms, each with an s and a p orbital shown, arc numbered as type I or 3 those which would be numbered 2 arc displaced by (I from the plane of the figure. 

Now that we know how to evolve the dynamics within a small time segment, we can decide to construct a Monte-Carlo-style stochastic algorithm to account for the quantum transitions that arise from the action of J. At the end of each time segment, the system either may remain in the same pair of adiabatic states or make a transition to a new pair of states. More specifically, for an initial pair of quantum states, (ckockq), the phase point R, P) is evolved for a time At to a new value RAt, PaO (here we use a simplified notation for the time-evolved phase points in the interval At) using the classical propagator and the phase factor is computed. With probability 1/2,... 

The real and imaginary parts of the complex propagation constant are called attenuation factor (Np m ) and the phase factor (Rad m ), respectively. They are given by Eqs (64) and (65) ... 

In the lagrangian formulation of quantum theory. S only enters as a phase factor in propagators, but disappears from squared modulus expressions such as transition probabilities [9], This already suggests the invariance of the physical predictions of the theory against different choices of the field G(x). One can also see this directly in the Hamiltonian formulation of quantum theory. If F is hermitian with respect to some inner product. 

At microwave frequencies (/ > 10 Hz) a polymer material is regarded as a distributed circuit having a complex propagation coefficient y = a ifi, where a and p are the electrical attenuation and phase factors of the material. These are measured using microwave techniques and e and e" follow as... 

The time evolution of the electronic eigenfunctions cpi obtained at time t = 0 from a quantum chemical code is given by their phase factor Q- dR(. )) xhe corresponding eigenenergies Ei(R(t)) are functions of the nuclear coordinates due to the propagating nuclear wavefunctions. To keep track of the actual phase factor of the electronic wavefunction from time step to time step this phase factor needs to be calculated recursively by utilizing the phase of the previous time step ... 

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