HermitePolynomials

1. The Hermite polynomials $H_n (x)$ are eigenfunctions of the Fourier transform operator $\mathcal{F}$, satisfying: $$\mathcal{F} [H_n (x)] = (i^n) \cdot H_n (k)$$ where $i$ is the imaginary unit, $n$ is the order of the polynomial, and $k$ is the frequency domain variable. The eigenvalues are $(i^n)$.

2. For the quantum harmonic oscillator, the position space wave functions are given by: $$\psi_n (x) = \frac{1}{\sqrt{2^n \cdot n! \cdot \sqrt{\pi}}} \cdot H_n (x) \cdot e^{- x^2 / 2}$$ Taking the Fourier transform, the momentum space wave functions are: $$\phi_n (p) = (i^n) \cdot \frac{1}{\sqrt{2^n \cdot n! \cdot \sqrt{\pi}}} \cdot H_n (p) \cdot e^{- p^2 / 2}$$ The Hermite polynomials enable an elegant transition between position and momentum space.

3. The kernel $K (x, x')$ of the integral form of the Schrödinger equation $$\psi (x) = \int K (x, x') \cdot \psi (x') dx'$$ is related to the Green's function $G (x, x' ; E)$ of the Schrödinger equation by: $$K (x, x') = \sum_n \psi_n (x) \cdot \psi^{\ast}_n (x') = \int G (x, x' ; E) \cdot dE$$ where $E$ is the energy. The Green's function satisfies: $$- \frac{\hbar^2}{2 m} \cdot \frac{d^2 G}{dx^2} + V (x) \cdot G (x, x' ; E) - E \cdot G (x, x' ; E) = \delta (x - x')$$ 4. The Green's function can be viewed as the derivative of the kernel $K$ with respect to the energy parameter $E$, treated as a Lebesgue-Stieltjes measure: $$\frac{dK (x, x')}{dE} = G (x, x' ; E)$$ This reflects how the Green's function filters the contribution to the propagation amplitude from a specific energy.

In conclusion, the Fourier transform properties of the Hermite polynomials play a crucial role in connecting the position and momentum space pictures in the quantum harmonic oscillator problem. The kernel and Green's function provide integral equation and source function perspectives on the Schrödinger equation and are also intimately related through the energy spectrum. These connections highlight the rich mathematical structure underlying this fundamental quantum system.