Hermite Functions: All You Need to Know
Part 4 of the mini-series on special functions, featuring a cool operator method for solving differential equations
No, that’s not a typo. These functions don’t live alone but they were discovered by a French guy called Charles Hermite, who is also eponymous for matrices that are called Hermitian. Ok, what’s special about Hermitian functions and why are they important? Every physics undergraduate will inevitably encounter them in quantum mechanics because they are — for example — the stationary state solutions for the wave functions of the quantum harmonic oscillator. And as such, they form an orthonormal basis of function space (well, actually only for square-integrable functions, but anyways). Beside of the usefulness of knowing about the Hermite functions per se, there is a cool operator method that can be used to solve their underlying differential equation. We’ll have a closer look at all that in the next sections.
The Hermite differential equation for the Hermite function 𝑦_𝑛(𝑥) reads
