HermiteWavePackets.jl

HermiteWavePackets.jl is a Julia package designed to handle elementary computations involving Hermite and Gaussian-type functions. The package supports the following classes of functions:

1. Real Gaussian Polynomials
   Functions of the form:

    P(x) e^{-\frac{a}{2} (x - q)^2},
   

   where:

   • $a \in \mathbb R_+^*$,
   • $q \in \mathbb R$,
   • $P(x)$ is a polynomial.

2. Complex Gaussian Wave Packets
   Functions of the form:

   P(x) e^{-\frac{z}{2} (x - q)^2} e^{i p x},
   

   where:

   • $z \in \mathbb C$ is a complex parameter (typically with (\text{Re}(z) > 0) to ensure decay),
   • $q, p \in \mathbb{R}$ are real-valued parameters representing spatial shift and momentum, respectively,
   • $P(x)$ is a polynomial.

The package is designed to provide efficient and accurate tools for computations involving these functions, which appear frequently in quantum mechanics, signal processing, and mathematical analysis.