Hermitefunktioille

Hermitefunktioille refers to Hermite functions, which are a set of orthogonal polynomials that appear in quantum mechanics and probability theory. They are solutions to the Hermite differential equation, a second-order linear ordinary differential equation. There are two common conventions for Hermite functions: the probabilistic and the physical ones. The probabilistic Hermite polynomials are defined using the error function and the Gaussian function. The physical Hermite polynomials are related to the probabilistic ones by a simple scaling factor. These functions are orthogonal with respect to a weight function, which is typically the Gaussian function. This orthogonality property makes them useful for expanding other functions as series. In quantum mechanics, they arise as the eigenfunctions of the quantum harmonic oscillator. In probability theory, they are related to the moments of the standard normal distribution. Their connection to the Fourier transform and other mathematical concepts further highlights their importance in various scientific fields.