The Husimi distribution for the eigenstates of a classically nonintegrable two-dimensional system with mixed phase space was calculated in order to analyze its structure in terms of classical trajectories. Besides eigenfunctions concentrated on invariant tori or with support on chaotic regions of phase space, we also found others that combine different classical invariant sets (like a stable and an unstable periodic orbit) and families of eigenstates scarred by classically unstable periodic orbits.
In this thesis, we study phase-space approximations to quantum dynamics for both Hermitian and non-Hermitian quantum systems, motivated by the need to efficiently describe quantum effects in models that are analytically intractable but admit classical analogues offering insight into their complex behaviour. Such systems arise in diverse areas, include but are not limited to quantum optics, condensed matter, open quantum systems, and quantum technologies. We begin by exploring phase-space approximations for quantum systems governed by the Heisenberg-Weyl algebra. Building on established coherent-state methods, we develop a semiclassical Husimi approximation for non-Hermitian Hamiltonians. This approach reveals that the resulting dynamics comprise the transport of initial Husimi distributions along classical trajectories, overlaid with a time-dependent norm landscape reflecting non-unitary evolution. We demonstrate the effectiveness of this method through several examples, including damped Kerr oscillators and
PT-symmetric models, providing a superior approximation for observables compared to standard coherent-state techniques. To generalise these ideas, we develop a unified framework for phase-space approximations that applies across a wide range of algebraic structures. By identifying a set of general criteria, we enable systematic application of the approximation scheme to any quantum system satisfying these conditions. This provides a cohesive theoretical foundation that connects previously distinct results under a common approach. We then apply this framework to both the
su2 algebra and non-Lie algebraic bosonic conversion systems. For
su2, we analyse two-mode Bose-Hubbard models with various Hamiltonians, demonstrating quantum-classical correspondence and exploring the effects of
PT-symmetry. For bosonic conversion systems, we propose coherent states for these non-Lie algebraic structures and derive their associated phase-space geometry, characterised as quantum Kummer shapes with classical counterparts as orbifolds. Our results advance the understanding of quantum-classical correspondence in multi-boson systems beyond traditional Hermitian frameworks and offer practical tools for simulating quantum dynamics in experimentally relevant models.
We construct a representation of the coherent state path integral using the Weyl symbol of the Hamiltonian operator. This representation is very different from the usual path integral forms suggested by Klauder and Skagerstam (1985 Coherent States: Applications in Physics and Mathematical Physics (Singapore: World Scientific)), which involve the normal or the antinormal ordering of the Hamiltonian. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. We show that the semiclassical limit of the coherent state propagator in the Weyl representation involves classical trajectories that are independent of the width of coherent states. This propagator is also free from the phase corrections found in Baranger et al ( 2001 J. Phys. A: Math. Gen. 34 7227 ) for the two Klauder forms and provides an explicit connection between the Wigner and the Husimi representations of the evolution operator.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
We present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are (finite-dimensional) Lie groups. The family of distributions on a phase space is postulated to satisfy the Stratonovich - Weyl correspondence with a generalized traciality condition. The corresponding family of the Stratonovich - Weyl kernels is constructed explicitly. In the presented theory we use the concept of generalized coherent states, that brings physical insight into the mathematical formalism.
The dynamics generated by non-Hermitian Hamiltonians are often less intuitive than those of conventional Hermitian systems. Even for models as simple as a complexified harmonic oscillator, the dynamics for generic initial states shows surprising features. Here we analyze the dynamics of the Husimi distribution in a semiclassical limit, illuminating the foundations of the full quantum evolution. The classical Husimi evolution is composed of two factors: (i) the initial Husimi distribution evaluated along phase-space trajectories and (ii) the final value of the norm corresponding to each phase-space point. Both factors conspire to lead to intriguing dynamical behaviors. We demonstrate how the full quantum dynamics unfolds on top of the classical Husimi dynamics for two instructive examples.