Quantum Harmonic oscillator

The eigenfunction of the quantum harmonic oscillator can be expressed in terms of the Hermite polynomials. The $n$-th eigenfunction $\Psi_n(x)$ is given by:

$$\Psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right) H_n\left(\sqrt{\frac{m\omega}{\hbar}} x\right)$$

where:

- $n$ is the quantum number, which takes on values of 0, 1, 2, \ldots

- $m$ is the mass of the particle,

- $\omega$ is the angular frequency of the oscillator,

- $\hbar$ (h-bar) is the reduced Planck's constant ($\hbar = h / 2\pi$, where $h$ is Planck's constant),

- $H_n(x)$ is the $n$-th Hermite polynomial,

- The factor $\frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}$ normalizes the wave function,

- The factor $\exp\left(-\frac{m\omega x^2}{2\hbar}\right)$ is a Gaussian envelope,

- The function $\sqrt{\frac{m\omega}{\hbar}} x$ is a scaling of the $x$ coordinate.

The Hermite polynomials are a set of orthogonal polynomials that appear frequently in quantum mechanics. They are defined through the generating function:

$$g(t, x) = \exp(-t^2 + 2tx) = \sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n$$

where the sum is over all nonnegative integers $n$. The $n$-th Hermite polynomial $H_n(x)$ is then given by the coefficient of $t^n$ in this series expansion. These polynomials can be computed explicitly using the recurrence relation:

$$H_0(x) = 1,$$

$$H_1(x) = 2x,$$

$$H_n(x) = 2x H_{n-1}(x) - 2(n-1) H_{n-2}(x) \quad \text{for} \quad n \geq 2.$$