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 > 0 ) is a positive real number,
   • ( q \in \mathbb{R} ) is a real-valued shift parameter,
   • ( 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.