Amplitude densities

INTRODUCTION DENSITY MATRIX TREATMENT Equation of motion for the density operator Variational method for the density amplitudes THE EIKONAL REPRESENTATION The eikonal representation for nuclear motions... 

This contribution considers systems which can be described with just the Hamiltonian, and do not need a dissipative term so that TZd = 0- This would be the case for an isolated system, or in phenomena where the dissipation effects can be represented by an additional operator to form a new effective non-Hermitian Hamiltonian. These will be called here Hamiltonian systems. For isolated systems with a Hermitian Hamiltonian, the normalization is constant over time and the density operator may be constructed in a simpler way. In effect, the initial operator may be expanded in its orthonormal eigenstates (density amplitudes) and eigenvalues Wn (positive populations), where n labels the states, in the form... 

Although the density amplitudes satisfy the standard Schroedinger differential equation for quantal states, they can depend on initial statistical conditions and are more general than standard states. 

The density amplitudes can usually be calculated more efficiently than the density operator because they depend on only one set of variables in a given representation although there are cases, such as shown below for the time-dependent Hartree-Fock density operator, where the advantages disappear and it is convenient to calculate the density operator. Expectation values of operators A t) follow from the trace over the density operator, as... 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

After solving this equation for the density matrix, its diagonalization provides the matrices of coefficients and the weights needed to reconstruct orthonormal density amplitudes. The density operator follows from r(f) = (<))r(f)( (f) ... 

Here a quantal potential has been introduced which implicitly depends on density amplitudes and their first and second derivatives with respect to Q, as... 

Results obtained from Eq.(22) and numerical solution of the RDM equation with the instantaneous dissipation from the dissipative potential formula, are in very good agreement with our previous results [29] using propagation of density amplitudes. The adsorbate state populations P/ reach at long times constant values, with Pg(oo) + Pe(oo) = 1 — Ps(oo) and Ps equal to the total population of the substrate, maintained by a steady interaction between p-and s-regions. 

Equation 37 has been used in an attempt130 to describe internal flexibility of the three hydroxymethyl groups of sucrose molecule in DzO solutions. The experimental data showed that the contribution of the overall motion to the spectral-density function of the hydroxymethyl group is similar to that of the ring carbons of sucrose. However, the presence of rapid internal motions about the three exocyclic bonds reduces the spectral density amplitudes. On the basis of the calculated order parameters in conjunction with model calculations, it was suggested130 that internal motions may be described as torsional librations. 

Differential Equation for Density Amplitude (p(r) and Concept of Pauli Potential... 

The wave function 1 for the ground state will now be written as the density amplitude p(r) already introduced in Sect. 6.2 times a phase factor exp(i0). It is useful for what is to follow to introduce the current density j at this stage (see also Appendix B). This is given, in the presence of the magnetic field, by... 

If the electron density of a crystal could be accurately described by a single cosine wave that repeats three times in the unit cell dimension, d, that is, has a periodicity of d/S, its diffraction pattern would have intensity only in the third order (only one Bragg reflection, 3 0 0). Conversely, if only one order of the diffraction spectrum is observed (the Bragg reflection h = 3, for example), then the diffracting density amplitude must correspond to a cosine wave with frequency d/h [dl3) (29, 30). This can be considered as an electron-density wave, one of the components summed to give an electron density map. Each Bragg reflection provides an electron-density wave that contributes to Equation 2, the total electron density. In this way the relationship between the order (hkl) of a Bragg reflection and its contribution to the electron density is established. 

Volume-phase grating Population (species)-phase grating Density-amplitude grating Temperature-amplitude grating Volume-amplitude grating Population (species)-amplitude grating... 

Figure 8.3 Spontaneous photon emission in a parabolic cavity The two-level system is positioned in the focus F of a metallic parabola with focal length f. All light rays emanating from F which are reflected at the parabolic boundary leave the cavity by propagating parallel to the symmetry axis. They all accumulate the same phase (eikonal) of magnitude + fj/c which is the same as if these light rays had started in phase from the plane z = —f. There are always two possible trajectories to any point x inside the cavity. In the semiclassical limit, i.e., f c/(Ueg. these two classes of trajectories give rise to the spherical-wave and the plane-wave contributions to the complex-valued energy-density amplitude of Eq. (32).

The equation of motion for Av5 which involves only a commutator with a Hamiltonian, could be solved by expanding the DOp in terms of density amplitudes satisfying a Schrodinger-like equation. More generally, however, the equation for a reduced DOp would contain dissipative rates, and this would make it necessary to solve the equation directly for the DOp. We therefore develop the numerical propagation method for the general case. The computational procedure starts with a basis set of quantum states, arranged as a row matrix l3>) = l< )2,...], taken here to be... 

Figure 5.11 shows Cu deposits obtained with the current density amplitudes of 0.20 A cm (Fig. 5.11a) and 0.44 A cm (Fig. 5.11b). In both cases, a deposition pulse of 1 ms and a pause duration of 10 ms were applied [22,23,55,57]. Formation of these deposits was accompanied by the quantity of evolved hydrogen which corresponded to the average current efficiency of hydrogen evolution, /i,av(H2), of 5.5 % with the applied current density amplitude, ic, of 0.20 A cm [57], and...

The increase of the current density amplitude and keeping durations of both the deposition pulse and pause constant... 

The prolonging of a deposition pulse duration and keeping both the current density amplitude and pause duration constant... 

The prolongation of a duration of deposition pulse leads to the formation of honeycomb-like structures with both the current density amplitudes applied [30, 45]. The formation of this structure type is analyzed applying the current density amplimde of 0.44 A/cm, deposition pulses of 1, 4, 7, 10, and 20 ms, and a pause duration of 10 ms [30]. The values of the average current efficiency of hydrogen evolution, 7i,av(H2), obtained for these parameters of PC regimes are... 

Fig. 4.19 Copper deposits obtained by the regime of pulsating current. The current density amplitude (a) 0.20 A/cm and (b) 0.44 A/cm. Deposition pulse 1 ms. Pause duration 10 ms (Reprinted from [30, 45] with permission from Elsevier.)...

Fig. 4.25 Copper grains agglomerates obtained by the PC regime with a current density amplitude of 0.44 A/cm, a deposition pulse of 20 ms, and a pause duration of 10 ms (Reprinted from [44] with pmnission frran Springer.)...

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