Comment on "Semiclassical and Quantum Analysis of a Focussing Free Particle Hermite Wavefunction", by Paul Strange (arXiv:1309.6753 [quant-ph])

TL;DR

This paper clarifies that a recent study on a free particle wavefunction is actually a Hermite-Gauss solution of the paraxial wave equation, showing that the described exotic solution is part of a well-known class of solutions.

Contribution

It demonstrates that the so-called exotic free particle wavefunction is a standard Hermite-Gauss solution, providing insight into the nature of such solutions and extending to Laguerre-Gauss examples.

Findings

The wavefunction is a Hermite-Gauss solution of the paraxial wave equation.

Exotic solutions can be represented by Laguerre-Gauss modes.

Clarifies the mathematical nature of the studied wavefunction.

Abstract

In the recent and very enjoyable paper (Paul Strange, "Semiclassical and Quantum Analysis of a Focussing Free Particle Hermite Wavefunction", arXiv:1309.6753[quant-ph]), Professor Strange has studied a particular solution of the free particle Schroedinger equation in which the time and space dependence are not separable. After recognizing the fact that "The Schroedinger equation has an identical mathematical form to the paraxial wave equation [...]", he claims to "describe and try to gain insight into an exotic, apparently accelerating solution of the free particle Schroedinger equation that is square integrable and which also displays some unusual characteristics." It is the main aim of this short note to show that the wavefunction described by Prof. Strange is simply one particular Hermite-Gauss solution of the paraxial wave equation and even more "exotic" examples can be found by considering, e.g., Laguerre-Gauss solutions.