Hermite Polynomials

Hermite polynomials (physicists' convention) $H_n(x)$ appear as the eigenfunctions of the quantum harmonic oscillator. They satisfy the differential equation:

$H_n''(x) - 2x\, H_n'(x) + 2n\, H_n(x) = 0$

Three-Term Recurrence

The most efficient way to evaluate $H_n(x)$ is via the recurrence relation:

$H_0(x) = 1, \quad H_1(x) = 2x$

$H_{n+1}(x) = 2x\, H_n(x) - 2n\, H_{n-1}(x)$

The first few polynomials are:

Quantum Harmonic Oscillator

The normalized energy eigenstates of the quantum harmonic oscillator (in dimensionless units) are:

$\psi_n(x) = \frac{1}{\sqrt{2^n\, n!\, \sqrt{\pi}}}\, H_n(x)\, e^{-x^2/2}$

These satisfy $\int_{-\infty}^{\infty} |\psi_n(x)|^2\, dx = 1$.

The probability density of finding the particle at position $x$ in state $n$ is:

$P_n(x) = |\psi_n(x)|^2 = \frac{H_n(x)^2\, e^{-x^2}}{2^n\, n!\, \sqrt{\pi}}$

Your Task

Implement hermite_H(n, x) using the three-term recurrence. Implement qho_wavefunction(n, x) for the normalized wavefunction $\psi_n(x)$. Implement qho_probability(n, x) for the probability density $|\psi_n(x)|^2$.

All constants (factorials, normalization, etc.) must be computed inside the function bodies.