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\begin{document}

\title{The Propagation of Hermite-Gauss wavepackets in Optics and Quantum
Mechanics}



\author[1]{John Briggs}%
\affil[1]{University of Freiburg}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\begin{abstract}
The two-dimensional paraxial equation of optics and the twodimensional
time-dependent Schr odinger equation, derived as approximations of the
three-dimensional Helmholtz equation and the three-dimensional
time-independent Schr odinger equation respectively, are identical. Here
the free propagation in space and time of Hermite-Gauss wavepackets
(optics) or Harmonic Oscillator eigenfunctions (quantum mechanics) is
examined in detail. The Gouy phase is shown to be a dynamic phase,
appearing as the integral of the adiabatic eigenfrequency or
eigenenergy. The wave packets propagate adiabatically in that at each
space or time point they are solutions of the instantaneous harmonic
problem. In both cases, it is shown that the form of the wave function
is unchanged along the loci of the normals to wave fronts. This
invariance along such trajectories is connected to the propagation of
the invariant amplitude of the corresponding free wave number (optics)
or momentum (quantum mechanics) wavepackets. It is shown that the van
Vleck classical density of trajectories function appears in the wave
function amplitude over the complete trajectory. A transformation to the
co-moving frame along a trajectory gives a constant wave function
multiplied by a simple energy or frequency phase factor. The Gouy phase
becomes the proper time in this frame.%
\end{abstract}%



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