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Form-preserving transformations of wave and Wigner functions

Mustafa Amin, Mason Daub, Mark A. Walton

TL;DR

The paper studies form-preserving transformations that map time-dependent Schrödinger solutions to solutions in possibly different potentials through time-dependent space-time reparameterizations. It derives the general 1D and D-dimensional transformations, including translations, scalings, and (when a vector potential is present) rotations, and shows how the transformed potentials and gauges relate via explicit formulas. In phase space, it shows the Wigner function transforms covariantly under the same canonical maps, explaining rigid evolutions of phase-space curves like those of Airy beams and coherent excited states, and it analyzes the Moyal equation to recover the same transformations. The results reveal a rich symmetry structure of nonrelativistic quantum mechanics with both scalar and vector potentials and connect wave-function dynamics to phase-space formulations, with potential links to gravitational settings and broader symmetry analyses.

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.

Form-preserving transformations of wave and Wigner functions

TL;DR

The paper studies form-preserving transformations that map time-dependent Schrödinger solutions to solutions in possibly different potentials through time-dependent space-time reparameterizations. It derives the general 1D and D-dimensional transformations, including translations, scalings, and (when a vector potential is present) rotations, and shows how the transformed potentials and gauges relate via explicit formulas. In phase space, it shows the Wigner function transforms covariantly under the same canonical maps, explaining rigid evolutions of phase-space curves like those of Airy beams and coherent excited states, and it analyzes the Moyal equation to recover the same transformations. The results reveal a rich symmetry structure of nonrelativistic quantum mechanics with both scalar and vector potentials and connect wave-function dynamics to phase-space formulations, with potential links to gravitational settings and broader symmetry analyses.

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 - 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.
Paper Structure (8 sections, 137 equations, 3 figures)

This paper contains 8 sections, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Remarkable time-dependent wave functions generated by form-preserving transformations
  3. Schrödinger form preservation in D space dimensions
  4. Schrödinger form preservation in 3 dimensions
  5. Form-preserving transformations in phase space
  6. Wigner function transformation from Schrödinger form preservation
  7. Form-preserving transformations of the Moyal equation
  8. Conclusion

Figures (3)

  • Figure 1: A phase-space parabola evolved rigidly in time by the free Hamiltonian. It relates to the free-space accelerating Airy beam solution of Berry and Balazs BB79, being a level curve (contour line) of the corresponding Wigner function MA19.
  • Figure 2: A phase-space circle evolved rigidly in time by the simple harmonic Hamiltonian. The Senitzky solutions describing coherent excited states Se54 have Wigner functions with level curves that behave this way.
  • Figure 3: A phase-space circle evolved in time by the free Hamiltonian, indicative of the free dispersion of harmonic-oscillator energy eigenstates.