Hermite Polynomials (Overview)

The quantum harmonic oscillator has energy eigenstates $\psi_n(x)$ that can be written in closed form using a family of polynomials first studied in the nineteenth century. Before writing those states down, it is worth introducing the polynomials themselves.

Physicist's Hermite Polynomials

The physicist's Hermite polynomial of degree $n$ is defined by Rodrigues' formula:

$H_n(u) = (-1)^n e^{u^2} \frac{d^n}{du^n} e^{-u^2}.$

The first several values are:

$H_0(u) = 1, \quad H_1(u) = 2u, \quad H_2(u) = 4u^2 - 2, \quad H_3(u) = 8u^3 - 12u.$

Each $H_n$ is an even function of $u$ when $n$ is even and an odd function when $n$ is odd — this parity follows directly from the Rodrigues formula because $e^{-u^2}$ is even and each differentiation flips the parity.

A useful three-term recurrence connects consecutive members of the family:

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

Starting from $H_0 = 1$ and $H_1 = 2u$ , this recurrence generates all higher-order polynomials without repeated use of the Rodrigues formula.

The Dimensionless Coordinate

For the quantum harmonic oscillator with mass $m$ and angular frequency $\omega$ , the natural length scale is

$\ell = \sqrt{\frac{\hbar}{m\omega}}.$

The dimensionless variable $u = x/\ell$ is what appears in the Hermite polynomials of the eigenfunctions. Working in $u$ keeps the expressions clean and makes the polynomial structure transparent.

Harmonic Oscillator Eigenfunctions

The normalized energy eigenfunction for the $n$ -th level is

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

The Gaussian factor $e^{-x^2/(2\ell^2)}$ ensures that every eigenfunction is square-integrable. The Hermite polynomial modulates that Gaussian: it introduces exactly $n$ nodes (zeros) into the wave function, which is consistent with the general rule that the $n$ -th excited state of any one-dimensional potential has exactly $n$ nodes.

Orthogonality and Normalization

The Hermite polynomials satisfy a weighted orthogonality relation:

$\int_{-\infty}^{\infty} H_m(u)\,H_n(u)\,e^{-u^2}\,du = \sqrt{\pi}\,2^n n!\,\delta_{mn}.$

The weight function $e^{-u^2}$ is precisely the squared Gaussian from the eigenfunctions. This means the eigenfunctions $\psi_n$ are automatically orthogonal:

$\int_{-\infty}^{\infty} \psi_m^*(x)\,\psi_n(x)\,dx = \delta_{mn}.$

Orthogonality is not a coincidence — it follows from the fact that $\psi_m$ and $\psi_n$ belong to different eigenvalues of the Hermitian Hamiltonian, so they must be orthogonal.

Recognizing Eigenfunctions in Practice

Given a wave function written as a product of a Gaussian and a polynomial, you can identify it as a harmonic oscillator eigenfunction by:

Checking that the exponent in the Gaussian has the form $-x^2/(2\ell^2)$ (or equivalently $-u^2/2$ ).
Matching the polynomial against the table of $H_n$ values above.
Verifying the prefactor using $1/\sqrt{2^n n!}$ and the normalization constant $(m\omega/\pi\hbar)^{1/4}$ .

For example, a wave function proportional to $(4u^2 - 2)\,e^{-u^2/2}$ matches $H_2(u)\,e^{-u^2/2}$ , so it is the $n = 2$ eigenstate. Its energy is $E_2 = (2 + 1/2)\hbar\omega = 5\hbar\omega/2$ .