Introduction to the Semiclassical Approximation

K Tomatani, José Luis Romero, A. B. Klimov

The semiclassical evolution of the generalized SU (2) Wigner function for quantum systems with a variable number of excitations is discussed in detail. It is shown that in the framework of the Liouvillian approach quantum dynamics can be approximately described in terms of classical trajectories on an appropriate four-dimensional symplectic manifold. The analysis of this manifold allows the introduction of a phase-like observable that can be interpreted as a relative phase ‘between’ invariant subspaces.

Joachim Ankerhold

Gert‐Ludwig Ingold

Introduction Path Integrals - Introduction - Propagator - Free Particle - Path Integral Representation of Quantum Mechanics - Particle on a Ring - Particle in a Box - Driven Harmonic Oscillator - Semiclassical Approximation - Imaginary Time Path Integral Dissipative Systems - Introduction - Environment as Collection of Harmonic Oscillators - Effective Action Damped Harmonic Oscillator - Partition Function - Ground State Energy and Density of States - Position Autocorrelation Function

August Burg

In this Master's thesis, we give a pedagogical introduction to the semiclassical analysis of wave dynamics in single-band and multi-band systems. We show how the semiclassical evolutions of Wigner functions can approximate the dynamics of wave packet centers and wave packet deformations. Three systems of increasing complexity are considered: the quantum harmonic oscillator and the quantum anharmonic oscillator as single-band systems, and the Dirac oscillator as a two-band system. We explain the origins of the first-order semiclassical corrections (Moyal, Littlejohn-Flynn, and Berry) in the equations of motion and examine their effects on predictive accuracy. Another focus is set on the conditions for validity of the semiclassical analysis. For both the quantum anharmonic and the Dirac oscillators, a dimensionless parameter controlling the accuracy of the approximation is identified. In multi-band systems, the formalism applies only to states within a single adiabatic subspace, which, for the Dirac oscillator, corresponds to an energy branch. This motivates the construction of single-branch almost coherent states: approximately Gaussian wave packets formed by superpositions of states from a single energy branch. Their centers undergo oscillations with an aperiodic envelope. These states enable comparison between semiclassical Wigner function evolution and the commonly used Sundaram-Niu formalism. We show that semiclassical Wigner function evolution captures deformation-related dynamics, including the initial decay of oscillations, which the Sundaram-Niu formalism cannot describe by construction. Additionally, exact quantum evolution reveals partial relocalization of these single-branch almost coherent states, a phenomenon that cannot be captured by the semiclassical evolution.

Zhuangqi Cao, Cheng Yin