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arXiv:1101.2683

Visualizing quantum mechanics in phase space

Heiko Bauke, Noya Ruth Itzhak

Published Jan 11, 2011• 6 citations
arXivHTMLPDF

Abstract

We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave functions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features.

DOI
10.48550/arxiv.1101.2683

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We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave functions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features.
This chapter presents some of the features of the phase space picture. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture. Using the Wigner distribution function it is possible to perform a canonical transformation in quantum mechanics. The phase space picture is a useful candidate to illustrate the transition from classical to quantum mechanics. The Poisson brackets and canonical transformations are part of the effective formulation of classical mechanics. In the infinite height potential video sequences of the evolution of Wigner functions for a wave function of a wave packet and its mirror wave packet is constructed. The chapter also presents the application of the canonical transformations and phase space distribution function in the study of wave packet spread and coherent and squeezed states.
We present a geometrical way of understanding the dynamics of wavefunctions in a free space, using the phase-space formulation of quantum mechanics. By visualizing the Wigner function, the spreading, shearing, the so-called "negative probability flow" of wavefunctions, and the long-time asymptotic dispersion, are intuited visually. These results are not new, but previous derivations were analytical, whereas this paper presents elementary geometric arguments that are almost "proofs without words", and suitable for a first course in quantum mechanics.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ\hbarℏ-dependent ⋆\star⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
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.
This chapter presents Wigner’s approach to quantum mechanics, based on the Wigner function in phase space. It explains Wigner–Weyl quantization, which makes it possible to associate functions on phase space to wave functions and operators, and it develops the technology to do quantum mechanics in this formalism. This includes the star product, Moyal evolution, and star-eigenvalue equations. It also develops semiclassical methods in this formulation, and it has a section on Berry’s semiclassical formula for the Wigner function in one-dimensional systems.