Hermite Polynomials
A sequence of classical orthogonal polynomials satisfying Hermite's differential equation y'' - 2x y' + 2n y = 0.
Governing formula
H_n(x) = (-1)^n e^(x²) (d^n / dx^n) (e^(-x²))
SI unit
Dimensionless Orthogonal Polynomials
In depth
Orthogonal with respect to weighting function e^(-x²). Hermite polynomials form the spatial wave function solutions for the quantum harmonic oscillator.
Examples in the real world
Expressing quantum harmonic oscillator spatial wave functions ψ_n(x) ∝ H_n(x) e^(-x²/2).