Form-preserving transformations of wave and Wigner functions

Quantum Physics arXiv:2510.21949 (quant-ph) [Submitted on 24 Oct 2025] Title:Form-preserving transformations of wave and Wigner functions Authors:Mustafa Amin, Mason Daub, Mark A. Walton View a PDF of the paper titled Form-preserving transformations of wave and Wigner functions, by Mustafa Amin and 2 other authors View PDF HTML (experimental) Abstract:Solutions of the time-dependent Schrödinger equation are mapped to other solutions for a (possibly) different potential by so-called form-preserving transformations. These time-dependent transformations of the space and time coordinates can produce remarkable solutions with surprising properties. A classic example is the force-free accelerating Airy beam found by Berry and Balazs. We review the 1-dimensional form-preserving transformations and show that they also yield Senitzky coherent excited states and the free dispersion of any harmonic-oscillator stationary state. Form preservation of the $D$- and 3-dimensional Schrödinger equation with both a scalar and a vector potential is then considered. Time-dependent rotations may be included when a vector potential is present; we find a general transformation formula for this case. Quantum form-preserving maps are also considered in phase space. First, the wave-function transformation is shown to produce a simple result for Wigner functions: they transform as a true phase-space probability would. The explicit transformation formula explains and generalizes the rigid evolution of curves in phase space that characterize the Airy beam and the coherent excited states. Then we recover the known form-preserving transformations from the Moyal equation obeyed by Wigner functions. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) Cite as: arXiv:2510.21949 [quant-ph] (or arXiv:2510.21949v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2510.21949 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mustafa Amin [view email] [v1] Fri, 24 Oct 2025 18:24:00 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled Form-preserving transformations of wave and Wigner functions, by Mustafa Amin and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-10 Change to browse by: math math-ph math.MP References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)