Quantum close-coupled calculations are performed for a model two-state curve-crossing problem that was previously treated by a semiclassical analysis. Excellent agreement in the differential cross sections is obtained. The interference structure on the elastic cross sections can be physically interpreted by using only the classical deflection functions of the adiabatic and diabatic potential curves and the interferences between them.
Quantum close-coupled calculations are performed for a model two-state curve-crossing problem that was previously treated by a semiclassical analysis. Excellent agreement in the differential cross sections is obtained. The interference structure on the elastic cross sections can be physically interpreted by using only the classical deflection functions of the adiabatic and diabatic potential curves and the interferences between them.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.
Quantum-mechanical calculations of the differential cross section for the small-angle elastic scattering of heavy particles are carried out to establish more definitely the region of validity of the classical approximation. Four results are discussed: (1) The Massey—Mohr phase-shift formula corresponds to the Kennard small-angle scattering formula in the semiclassical limit. (2) The Schiff approximation for the cross section is exactly the same as the semiclassical approximation at small angles, for any central potential. (3) At very small angles the semiclassical limit for the differential cross section varies as exp (—cθ2), where c is a function of velocity for which explicit expressions are given. (4) The first quantum deviation from the classical limit, which is proportional to ℏ2, can be combined with the preceding result to give a reasonable representation of the differential cross section over the entire range of small angles for which quantum deviations are appreciable. Detailed calculations for some specific systems are made, and it is shown that Wu's misgivings over the classical interpretation of the experimental results of Amdur and co-workers are unjustified.
Abstract Semiclassical mechanics, which stems from the old quantum theory, has seen a remarkable revival in recent years as a physically intuitive and computationally accurate scheme for the interpretation of modern experiments. The main text concentrates less on the mathematical foundations than on the global influence of the classical phase space structures on the quantum mechanical observables. Further mathematical detail is contained in the appendices. Worked problem sets are included as an aid to the student. One important thread concerns the sharp separation between quantum interference in the classically accessible phase space and exponential decay into the inaccessible shadow regions. Another is the use of catastrophe theory to eliminate singularities at the classical boundaries between the two types of region. The theory goes on to show how different parts of the wavefunction may be stitched together to obtain accurate quantization formulae for a wide variety of tunnelling and curve-crossing problems. A careful account of classical angle–action transformations is extended to include the evaluation of matrix elements. A variety of extensions of the familiar spectroscopic RKR inversion procedure are given. It is also shown how bifurcations in the phase space structure signal changes in the quantum eigenvalue spectrum from, for example, normal mode to local mode behaviour. Other quantization techniques are shown to extend even to weakly chaotic systems. A chapter on Gaussian wavepacket propagation leads up to the currently popular Herman–Kluk technique. The book concludes with chapters on semiclassical aspects of elastic, inelastic and reactive scattering.
We use the adiabatic theory in the complex plane of internuclear distance to calculate the Rydberg-state broadening cross sections in the ambient alkali-metal gases. We have found that the imaginary parts of the branch points between the relevant energy curves have an oscillatory structure that is projected into a similar structure in the broadening cross section. The results of the distorted-wave diabatic perturbation theory and the present theory are in excellent agreement for the principal quantum numbers of the Rydberg atoms n30. Unlike the diabatic perturbation theory, the adiabatic approach does not merge with the impulse approximation and fails for higher n.
‘‘Multistructure’’ cross sections, of both conventional and coherence types, are calculated for the fine structure transition Na(2P1/2←2P3/2)+He in an energy range of 10–200 meV. The cross sections are related to conditions of a crossed beams experiment in which the three controlled vectors are the initial relative velocity and the two polarization directions of excitation and probe laser beams. Both semiclassical and quantum mechanical calculations are performed. The semiclassical method is employed to interpret and visualize the collision mechanism leading to the multistructure cross sections. In addition, the validity of the approximations used in the semiclassical model is investigated. For the conventional cross sections, the semiclassical and quantum-mechanical results are in good agreement. We find that both initial sodium states 3 2P3/2,3/2 and 3 2P3/2,1/2 preferentially populate the 3 2P1/2,−1/2 final state, i.e., a final state in which the sign of the magnetic quantum number has changed. Vector evolution diagrams are introduced to interpret the conventional cross sections and the pattern of the Stückelberg-like oscillations. The semiclassical model gives the correct order of magnitude for the coherence cross sections, but in this case, is inadequate in the quantitative description; thus the coherence cross sections are mainly studied with full quantum calculations. The coherence cross section is especially sensitive to small variations in the energy difference and shape of the potential curves. The value of the coherence cross section is strongly correlated to the position and well depth of the potential minimum of the 2Π state.
This series of papers is concerned with the derivation of the equations of the classical picture of atomic collisions, $i\ensuremath{\hbar}\frac{d}{\mathrm{dt}}{d}_{i}(t)=\ensuremath{\Sigma}\stackrel{}{j}{h}_{\mathrm{ij}}(t){d}_{j}(t),$ which describe the "time" dependence of electronic-quantum-state amplitudes as the nuclei move along a classical trajectory. These equations are derived in two ways. In the first formulation, which coincides with the intuitive classical picture of the collision, the nuclear part of the wave function is treated as a superposition of narrow wave packets, each traveling along a classical trajectory. In the second formulation, a semiclassical approach is used. The validity and meaning of the two formulations are discussed and compared.