Semiclassical Time Propagation and the Raman Spectrum of Periodic Systems

Abstract

The first half of this thesis introduces the time-dependent W.K.B. approximation of quantum mechanics from basic principles in classical and quantum mechanics. After discussing the van Vleck-Morette-Gutzwiller propagator, the real-trajectory time-dependent W.K.B. approximation of a coherent state is introduced. This is also called the off-center ''thawed'' Gaussian approximation and has a closed-form solution consisting of a Gaussian with time-dependent position and momentum, dispersion, and position-momentum correlation. This result is then extended to third order in the classical action of guiding real trajectories - a parabolization in phase space, and equivalently, a uniformization over two saddle points - allowing for the novel treatment of non-linearity in its underlying classical dynamics. The result is another simple closed-form solution, but this time made up of Airy functions and their derivatives multiplied by an exponential. Unlike the lower-order treatment, which stopped at linearization of phase space, this expression is able to capture global as well as local non-linear dynamics at finite Planck's constant.

We then proceed to discuss another uniformization of the semiclassical primitive propagator: the Heller-Herman-Kluk-Kay (H.H.K.K.) propagator. The H.H.K.K. involves an integral over all of phase space which can be trimmed down to only a one-dimensional integral, regardless of the dimensions of the system, by appealing to similar guiding manifold techniques discussed in the previous section. This is the basis for the directed H.H.K.K. propagator which we investigate. Though many possibilities for speeding up the semiclassical evaluation of H.H.K.K. been examined over the years, few have focused on using the actual dynamics of underlying trajectories to simplify its computation. Our findings offer encouraging evidence about the promise of this direction.

The second half of this thesis is concerned with describing the Raman spectrum of graphene within the Born-Oppenheimer approximation using the Kramers-Heisenberg-Dirac (K.H.D.) formalism. The electronic and vibrational properties of graphene are introduced, along with simple tight-binding methods of calculating them. With these tools, K.H.D. is then applied to explain the origin of the unique and few prominent peaks in graphene's Raman spectrum. Here, the dominant effect of graphene's linear Dirac cone in its electronic dispersion is easily seen. The latter leads to novel electron-light-phonon ''sliding transitions'' that explain the brightness of the overtone 2D peak.

Finally, some more minor results on the subject of the asymptotic zeros of orthogonal polynomials are presented.