Semiclassical scattering theory can be summarized as the study of connections between classical mechanics and quantum mechanics in the limit ℏ → 0 over the infinite time domain -∞ < t < ∞. After a brief discussion of Semiclassical Analysis and Scattering Theory we provide a rigorous result concerning the time propogation of a semiclassical wavepacket over the time domain -∞ < t < ∞. This result has long been known for dimension n ≥ 3, and we extend it to one and two space dimensions. Next, we present a brief mathematical discussion of the three body problem, first in classical mechanics and then in quantum mechanics. Finally using an approach similar to the semiclassical wave-packet construction we form a semiclassical approximation to the solution of the Schrödinger equation for the three-body problem over the time domain -∞ < t < ∞. This technique accounts for clustering at infinite times and should be applicable for researchers studying simple recombination problems from quantum chemistry. / Ph. D.

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<p> An expression for the semiclassical density of states for a particle in a smooth potential well is obtained from the Kirkwood expansion of the partition function. This expression for the semiclassical density of states is then shown to be essentially equivalent to the expression obtained from the Green's function method of Balian and Bloch.</p> <p> The Strutinsky shell correction to the nuclear binding energy is then analytically shown to be equivalent to the shell correction obtained from a consideration of the semiclassical partition function if certain restrictions on the Strutinsky smoothing parameter can be met.</p> / Thesis / Master of Science (MSc)

A quantum/semiclassical theory for the internal nuclear dynamics of a small molecule and the induced small-amplitude coherent motion of a low-temperature host medium is developed, tested and applied to simulate and interpret ultrafast optical signals. Linear wave-packet interferometry and time-resolved coherent anti-Stokes Raman scattering signals for a model of molecular iodine in a 2D krypton lattice are calculated and used to study the vibrational decoherence and energy dissipation of iodine molecules in condensed media. The total wave function of the whole model is approximately obtained instead of a reduced system density matrix, and therefore the theory enables us to analyze the behavior and the role of the host matrix in quantum dynamics. This dissertation includes previously published co-authored material.

Motivated by the desire to understand classical-quantum correspondences, we study concentration phenomena of approximate eigenfunctions of a semiclassical pseudodifferential operator $P(h)$. Such eigenfunctions appear as steady state solutions of quantum systems. Here we think of $h$ as being a small parameter such that $h^{2}$ is inversely proportional to the energy of such a system. As we understand classical mechanics to be the high energy (or small $h$) limit of quantum mechanics we expect the behaviour of eigenfunctions $u(h)$ for small $h$ to be related to properties of the associated classical system. In particular we study the connection between the classical flow and the quantum concentration properties. The flow, $(x(t),\xi(t))$, of a classical system describes the system's motion through phase space where $x(t)$ is interpreted as position and $\xi(t)$ is interpreted as momentum. In the quantum regime we think of an eigenfunction as being composed of highly localised packets moving along bicharacteristics of the classical flow. With this intuition we relate concentration of eigenfunctions in a region to the time spent by projections of bicharacteristics there. We use the $L^{p}$ norm of $u$ when restricted to submanifolds as a measure of concentration. A high $L^{p}$ norm particularly for small $p$ is indicative of concentration near the submanifold. We reduce the estimates on eigenfunctions to operator norm estimates on associated evolution operators. Using the semiclassical analysis methods developed in Chapter 3 we express these evolution operators as oscillatory integral operators. Chapter 2 covers the technical background needed to work with such operators. In Chapter 4 we determine eigenfunction estimates for eigenfunctions restricted to a smooth embedded submanifold $Y$ of arbitrary dimension. If $Y$ is a hypersurface, the greatest concentration occurs when there are bicharacteristics of the classical flow embedded in $Y$. In Chapter 5 we assume that projections of such bicharacteristics can be at worst simply tangent to $Y$ and thereby obtain better results for small values of $p$.

Semiclassical and Born-Oppenheimer approximations are used to provide uniform error bounds for the energies of diatomic molecules for bounded vibrational quantum number n and large angular momentum quantum number l. Specifically, results are given when (l + 1) < κ𝛜⁻³/². Explicit formulas for the approximate energies are also given. Numerical comparisons for the H+₂ and HD+ molecules are presented. / Ph. D.

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We investigate semiclassical contributions to correlation functions in N = 1 supersymmetric gauge theories. Our principal example is the gluino condensate, which signals the breaking of chiral symmetry, and should be exactly calculable, according to a persymmetric non-renormalisation theorem. However, the two calculational approaches previously employed, SCI and WCI methods, yield different values of the gluino condensate. We describe work undertaken to resolve this discrepancy, involving a new type of calculation in which the space is changed from R(^4) to the cylinder R(3) x S(1) This brings control over the coupling, and supersymmetry ensures that we are able to continue to large radii and extract answers relevant to R(^4). The dominant semiclassical configurations on the cylinder are all possible combinations of various types of fundamental monopoles. One specific combination is a periodic instanton, so monopoles are the analogue of the instanton partons that have been conjectured to be important at strong coupling. Other combinations provide significant contributions that are neglected in the SCI approach. Monopoles are shown to generate a superpotential that determines the quantum vacuum, where the theory is confining. The gluino condensate is calculated by summing the direct contributions from all fundamental monopoles. It is found to be in agreement with the WCI result for any classical gauge group, whereas the values for the exceptional groups have not been calculated before. The ADS superpotential, which describes the low energy dynamics of matter in a supersymmetric gauge theory, is derived using monopoles for all cases where instantons do not contribute. We report on progress made towards a two monopole calculation, in an attempt to quantify the missed contributions of the SCI method. Unfortunately, this eventually proved too complicated to be feasible.

The objective of the thesis is to treat the Schrodinger equation in parallel with a standard treatment of the heat equation. In the books of the Rubensteins and Kress, the heat equation initial value problem is converted into a Volterra integral equation of the second kind, and then the Picard algorithm is used to find the exact solution of the integral equation. Similarly, the Schrodinger equation boundary initial value problem can be turned into a Volterra integral equation. We follow the books of the Rubinsteins and Kress to show for the Schrodinger equation similar results to those for the heat equation. The thesis proves that the Schrodinger equation with a source function does indeed have a unique solution. The Poisson integral formula with the Schrodinger kernel is shown to hold in the Abel summable sense. The Green functions are introduced in order to obtain a representation for any function which satisfies the Schrodinger initial-boundary value problem. The Picard method of successive approximations is to be used to construct an approximate solution which should approach the exact Green function as n goes to infinity. To prove convergence, Volterra kernels are introduced in arbitrary Banach spaces, and the Volterra and General Volterra theorems are proved and used in order to show that the Neumann series for the L^1 kernel, the L^infinity kernel, the Hilbert-Schmidt kernel, the unitary kernel, and the WKB kernel converge to the exact Green function. In the WKB case, the solution of the Schrodinger equation is given in terms of classical paths; that is, the multiple scattering expansions are used to construct from, the action S, the quantum Green function. Then the interior Dirichlet problem is converted into a Volterra integral problem, and it is shown that Volterra integral equation with the quantum surface kernel can be solved by the method of successive approximations.