Transition amplitude

As seen from this table, the WKB approximation is reasonably accurate even for very shallow potentials. At 7 = 0 the hindered rotation is a coherent tunneling process like that studied in section 2.3 for the double well. If, for instance, the system is initially prepared in one of the wells, say, with cp = 0, then the probability to find it in one of the other wells is P( jn, t) = 5sin (2Ar), while the survival probability equals 1 — sin ( Ar). The transition amplitude A t), defined as P( + t) = A t), is connected with the tunneling frequency by... 

In the random phase approximation, the transition amplitude from state 0) to l) for any one electron operator O may be written as... 

In order to estimate the phonon scattering strength and thus the heat conductivity, we need to know the effective scattering density of states, the transition amplitudes, and the coupling of these transitions to the phonons. 

We do not possess detailed information on the transition amplitudes however, they should be on the order of the transition frequencies themselves, just as is the case for those two-level systems that are primarily responsible for the phonon... 

We now calculate the density of the phonon scattering states. Since we have effectively isolated the transition amplitude issue, the fact of equally strong coupling of all transitions to the lattice means that the scattering density should directly follow from the partition function of a domain via the... 

Now we try to control the transition by sweeping the field parameter F as a function of time striding the avoided crossing position F. First, let us consider n periods of oscillation between Fa and Ft,. The final overall transition amplitude Tn is given by... 

The parameters c and 8 are defined in Section A.3 The nonadiabatic transition amplitude that connects the wave function just before and right after the transition at the avoided crossing is given by... 

Tj and are the turning points on E2 R) (see Fig. Al). The nonadiabatic transition amplitude to connect the wave functions between the right and the left sides of the crossing point is given by... 

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

The detection probability for a given trajectory depends on the fragment orientation (its Mj value) and the nature of the probe transition. All of these images were obtained via the two-photon Ilg XAS) transition. Five rotational branches are thus possible O, P, Q, R and S. The amplitudes for each of these two-photon transitions can be obtained from a sum of paired, Mj-dependent, one-photon amplitudes.37 The O branch, for example, consists of a contribution from a parallel P-type transition to a 7A virtual state, followed by a perpendicular P-type transition to the final 1ffs Rydberg (which is assumed to be ionized promptly). The product of those two transition amplitudes must be summed with another product in which the first transition is perpendicular and the second is parallel. The P and R branches consist of four contributions each and the Q branch has six such terms in its transition amplitude. The required one-photon amplitudes are taken from Ref. 37. 

A particularly useful probe of remote-substituent influences is provided by optical rotatory dispersion (ORD),106 the frequency-dependent optical activity of chiral molecules. The quantum-mechanical theory of optical activity, as developed by Rosenfeld,107 establishes that the rotatory strength R0k ol a o —> k spectroscopic transition is proportional to the scalar product of electric dipole (/lei) and magnetic dipole (m,rag) transition amplitudes,... 

In chemical physics, having knowledge of just the eigenpairs of the relevant Hamiltonian is often insufficient because many processes involve transition between different states. In such cases, the transition amplitudes between these states under a quantum mechanical propagator may be required to solve the problem ... 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

Both RRGM and SLP have been used to compute various transition amplitudes with high efficiency and accuracy. Their applications, which have been reviewed in the literature,56 57 59 include laser-molecule interaction,43 44 99 correlation functions,45 104 absorption and emission spectra,100 103 105-107 intramolecular energy transfer,108-115 vibrational assignment,103 116 117 and reaction dynamics. ... 

It was demonstrated in the above subsection that the Lanczos algorithm can be used to compute scalar quantities such as transition amplitudes without explicit calculation and storage of the eigenvectors. We discuss here another... 

Transition Amplitudes with a Single Lanczos Propagation. 

Method for Calculating Transition Amplitudes. II. Modified QL and Symmetry Adaptation. 

To introduce the basic concept of a semiclassical propagator, let us consider an n-dimensional quantum system with Hamiltonian H, which is assumed to possess a well-defined classical analog. In order to obtain the semiclassical approximation to the transition amplitude between the initial... 

The evaluation of the semiclassical Van Vleck-Gutzwiller propagator (106) amounts to the solution of a boundary-value problem. That is, given a trajectory characterized by the position q(f) = q, and momentum p(f) = p, we need to hnd the roots of the equation = q floiPo)- To circumvent this cumbersome root search, one may rewrite the semiclassical expression for the transition amplitude (105) as an initial-value problem [104-111]... 

Kluk propagator, the semiclassical approximation for this transition amplitude is... 

To discuss the semiclassical spin-coherent state propagator, we consider a general transition amplitude ( / e which can be expressed as an integral... 

In the 1950s, many basic nuclear properties and phenomena were qualitatively understood in terms of single-particle and/or collective degrees of freedom. A hot topic was the study of collective excitations of nuclei such as giant dipole resonance or shape vibrations, and the state-of-the-art method was the nuclear shell model plus random phase approximation (RPA). With improved experimental precision and theoretical ambitions in the 1960s, the nuclear many-body problem was born. The importance of the ground-state correlations for the transition amplitudes to excited states was recognized. 

M. Rosina, Transition amplitudes as ground state variational parameters, in Reduced Density Matrices with Applications to Physical and Chemical Systems (A. J. Coleman and R. M. Erdahl, eds.), Queen s Papers in Pure and Applied Mathematics No. 11, Queen s University, Kingston, Ontario, 1967, p. 369. 

---