Introduction: The Special Functions of the Oscillator
The terminated power series solutions are Hermite polynomials \(H_n(\xi)\). These form a complete orthogonal set on \(\mathbb{R}\) with weight function \(e^{-\xi^2}\).
First Few Hermite Polynomials
- \(H_0(\xi) = 1\)
- \(H_1(\xi) = 2\xi\)
- \(H_2(\xi) = 4\xi^2 - 2\)
- \(H_3(\xi) = 8\xi^3 - 12\xi\)
- \(H_4(\xi) = 16\xi^4 - 48\xi^2 + 12\)
Generating Formula
\[H_n(\xi) = (-1)^n e^{\xi^2} \frac{d^n}{d\xi^n}e^{-\xi^2}\]The Wavefunctions
\[\psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n(\xi) e^{-\xi^2/2}\]where \(\xi = \sqrt{m\omega/\hbar}\,x\).
The Quantum Connection
Hermite polynomials appear throughout physics wherever harmonic motion is quantized. Their orthogonality ensures that different energy eigenstates are distinguishable. The parity of \(H_n\) is \((-1)^n\): even states are symmetric, odd states antisymmetric.