4.2 Hermite'sche Polynome

Energieeigenfunktionen des harmonischen Oszillators in der Quantenmechanik^4.1

$$\leq \leq \infty$$ (4.5)

$${1\over {2^n n! \sqrt{\pi}}} \infty \infty$$ (4.6)

H₀(x) = 1        H₁(x) = 2x        H₂(x) = 4x² - 2        H₃(x) = 8x³ - 12x (4.7)

allgemein:

$${d^n \over {dx^n}}$$ (4.8)

$${{(-1)^n(2n)!} \over n!}$$ (4.9)

Rekursionsformel zur Berechnung:

H_{n + 1}(x) = 2xH_n(x) - 2nH_{n - 1}(x) (4.10)

$$\left(\vphantom{{d^2 \over {dx^2}} -2x {d \over {dx}} + 2n }\right. {d^2 \over {dx^2}} {d \over {dx}} \left.\vphantom{{d^2 \over {dx^2}} -2x {d \over {dx}} + 2n }\right)$$ (4.11)