The behaviour of classical systems is characterised by their phase portraits; the collections of their trajectories. In quantum mechanics phase portraits are still considered impossible to compute due to the complexity of quantum trajectories arising from the introduction of the quantum correction terms. Instead, in this thesis, we identify the Wigner current (the rate of flow per unit area of the Wigner distribution), as the quantum analogue of the classical phase-space current, and through Wigner current's fieldline portraits we show that it reveals hidden features of quantum dynamics and extra complexity. In our analysis, we focus on the simplest, most intuitive, and analytically accessible aspects of the Wigner current. We investigate its features for weakly-anharmonic weakly-excited bound-states of time-reversible one-dimensional quantum-mechanical systems. We establish that weakly-anharmonic potentials can be grouped into three distinct classes: hard, soft, and odd. We stress connections between each other and the harmonic case. We show that their Wigner current fieldline portraits can be characterised by the Wigner current's discrete stagnation points, how these arise and how a quantum system's dynamics is constrained by the stagnation points' topological charge conservation. We additionally demonstrate the conceptual power of the Wigner current by addressing some confusion found in the literature. We also stress the usefulness of the integral form of Wigner's representation as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold starproducts. The associated integral form of the Wigner current is used here in an elementary proof which shows that only systems up to quadratic in their potential fulfil Liouville's theorem of volume preservation in quantum mechanics.
This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions density -- matrices in a special Weyl representation -- and star products are the cornerstones of the formalism. The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products. This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics. Readership: Advanced undergraduates and graduates studying quantum mechanics.
Phase space is the unity of position and momentum configuration space. It allows for an effective description of dynamical systems and is particularly useful when it comes to studying chaos theory and statistical mechanics. After the advent of quantum physics early in the 20th century, E. Wigner [91], J. E. Moyal [62] and H. J. Groenewold [31] introduce a quantum theory in phase space. Despite the apparent added complexity of the mathematics involved in this new framework, the underlying classical and quantum equations show many similarities. The probability distribution in classical physics becomes the Wigner distribution, a probability distribution usually featuring negative values. In 2013, O. Steuernagel and D. Kakofengitis, inspired by the work of H. Bauke [7] and E. Wigner [91], identified the quantum analogue of the classical phase space flow: the Wigner current J [83]. This Wigner current allows the visualisation of quantum dynamics through a quantum fluid dynamics perspective in phase space. This thesis is written by collection of five articles. They are prefaced by an introduction into the basics of quantum phase space theory and its link with both classical phase space dynamics and the standard Schrödinger approach, followed by the articles published during this PhD. Article 1 shows the importance of the integral form of the Wigner current. We use it to derive the Ehrenfest’s theorem, as well as to refute some propositions made within the community. Article 2 shows that, using the Wigner current, an Eulerian and Lagrangian point of view do not always give the same results for the quantum case. We demonstrate that the negativities of the Wigner distribution, sign of quantumness of the system, are created by the Wigner velocity field singularities. The Wigner velocity field is the quantum analogue of the classical phase space velocity field. In Article 3, we see that even though Wigner distributions of quantum systems feature spotty structures which saturate on scales ɑZ [97], the construction of a superoscillating Wigner distribution allows one to generate much smaller structures, of the order of ɑZ /α with α a positive constant potentially very large. In Article 4, we introduce the concept of quantum shear suppression in phase space. The Wigner current features an effective quantum “viscosity”, suppressing classical dynamics fine details. This viscosity is the mechanism by which the Zurek scale is enforced dynamically onto the state in phase space. In Article 5, we apply the previous ideas to Kerr-type oscillators. Its Wigner current is derived, and using it we show that its values are conserved on a ring during the time evolution of the Kerr oscillator. The shear suppression is also studied.
The dynamical evolution is described within the phase-space formalism by means of the Moyal propagator, which is the symbol of the evolution operator. Quadratic Hamiltonians on the phase space are distinguished in that their Moyal bracket with any function equals their Poisson bracket. It is shown that, for general time-independent quadratic Hamiltonians, the Moyal propagators transform covariantly under linear canonical transformations; they are then derived and classified in a fully explicit manner using the theory of Hamiltonian normal forms. The authors present several tables of propagators. It is proved that these propagators belong to the Moyal algebra of distributions, and that the spectrum of the Hamiltonian may be obtained directly as the support of the Fourier transform of the Moyal propagator with respect to time. From that, the quantum-mechanical problem for these Hamiltonians is, in principle, completely solved. The appropriate path-integral formalism for phase-space quantum mechanics, leading back to the same results, is outlined in an appendix.
This thesis addresses aspects of quantum phase and sets out to draw attention to a collection of new and extend some not so new techniques utilizing Weyl's correspondence rule and its inverse. The phase formulation of quantum mechanics is as old as quantum mechanics itself and evolved over a period of two decades due to the contributions of H. Weyl \cite{Weyl1950}, E. P. Wigner \cite{Wigner1932}, H. Groenewold \cite{Groenewold1946} and J. E. Moyal \cite{Moyal1949}. Over the ensuing years the phase picture has matured into a practical analytical tool, and has generated a multitude of technical papers, reports, reviews, and expositions from many authors and commentators. One reason for its success is that it offers an intuitive picture of quantum mechanics owing to its \it{classical look and feel}, and it has made an impression, in one way or another, in many fields such as mathematical physics, quantum optics, and physical chemistry.The thesis opens with a brief history of classical phase space, tracing the possible origin of the term ``phase space itself, and chronicles the events leading to the conceptualization (and generalization) of quantum phase space. In Chapter 2 the Weyl correspondence is recast in terms of von Neumann's integral kernel transform and a heuristic analysis is given from this perspective. A characteristic of Weyl's correspondence rule and its inverse is that there are two inherent representations of the same operator. These have been referred to as the Weyl representation and Wigner representation and are related by a symplectic Fourier transform. For an arbitrary outer dyadic operator it is shown that the Weyl representation leads to the \it{modified} Gabor transform (known to Signal Processing), while the Wigner representation specializes to the Bargmann transform and a representation in terms of analytic functions. Calculating Weyl-Wigner (WW) transforms of products of
n operators, from first principles, has proved to be a technically difficult problem. However, with the machinery developed in Chapter 2 the problem is reduced to a straightforward algebraic exercise. Finally, forming an
s-parametrized generalization of Weyl's correspondence rule and focusing on the Weyl representation it is shown to lead to an intimate relationship with the relative state formalism \cite{Ban1998}. Weyl's correspondence rule is only applicable to operators and consequently state vectors
∣ψ⟩ are not represented in the usual formulation of quantum phase space. This problem is circumvented in Chapter 3 by introducing the notion of a normalized fixed state (or window) vector
∣φ0⟩ to form an outer dyadic product $\wh{\Psi} = |\psi\rangle\langle\varphi_0|$. The WW transform of this product under the action of the Weyl's rule is a complex valued function of real variables called a phase amplitude and denoted by the symbol
Ψ. The phase amplitudes have many (and sometimes surprising) properties. For example they: \begin{itemize} \item form the building blocks of Wigner functions
W by admitting a generalized Born rule
W=Ψ⋆Ψˉ, and as such can be considered to be a more fundamental objects than Wigner functions; \item superpose and interfere like the state vectors they represent; \item specialize to the Bargmann and Gabor transforms when a Gaussian window is chosen; \end{itemize} \ni In Chapter 4 further aspects of phase amplitudes are explored and discussed. For example it is shown that they: \begin{itemize} \item define phase state vectors which form an over complete set and are elements of a reproducing kernel Hilbert space; \item allow unitary ray representations of the group symmetries of translation, rotation and time reversal; \item evolve in phase under the influence of a propagator
R which is intimately related to the Wigner propagator
P. \end{itemize} Chapter 5 considers an extension of phase amplitudes to include systems with finitely many degrees of freedom. These are called spin amplitudes and are obtained by first forming the outer dyadic product of a spin state vector with an arbitrary fixed spin state vector, and then applying a finite-dimensional analogue of the WW transform. The spin amplitude operator is represented as an
n×n matrix, whose elements are products of the spin components of the spin and fixed state vectors. Two kinds of spin amplitudes are considered, arising on the sphere
S2 or on a
(2j+1) dimensional lattice. In the special case of spin-$\half$ systems it is shown that the spin amplitudes transform as spinors under rotations on both the sphere and lattice. As a physical application the spin amplitudes for a spin-$\half$ magnetic dipole are found for a Nuclear Magnetic Resonance (NMR) environment. Chapter 6 considers Schwinger's abstract approach to quantum mechanics which takes into account the inability to indefinitely compensate for fluctuations in the measurement process. The resulting measurement algebra provides a symbolic representation of the physical laws as opposed to the mathematical abstraction (of von Neumann and Dirac) that was justified \it{\'a posteriori} by being in agreement with experiment. Despite Schwinger's prolific work on the subject of measurement algebra, its interpretation in quantum phase remained an open issue. It is shown that the key to formulating a quantum phase in terms of the measurement operators and transformation functions is the determination of a suitable set of basis operators for an arbitrary measurement operator. Many new results follow as a consequence of expanding an arbitrary measurement operator in this basis. In Chapter 7 the quantum-classical interface is explored by considering Hamilton's principal function or action
S(q,v,t). Starting from a purely heuristic classical argument, a phase function
U(p,q,t) is derived in terms of the classical action and is shown to be proportional to the WW transform of the quantum evolution operator. This association leads to the transform \be U(p,q,t) = A(t) \int e^{ipv/\hbar} e^{iS(q,v,t)/\hbar} \; \rm{d}v, \ee which enables one, in principle, to deduce the Wigner functions of a given system when only the classical action is known. Simple physical systems are used to provide examples illustrating this approach.