Hermite Polynomials and Oscillator Eigenfunctions

Point a beam profiler at a slightly misaligned laser and you will photograph Hermite polynomials: the transverse modes of a stable resonator are Hermite-Gaussians, labelled TEM$_{mn}$ by their node counts — the gallery above, glowing on a screen. In quantum optics the connection runs deeper: a field mode is an oscillator, its photon-number (Fock) states are exactly the $\psi_n$, and homodyne tomography reconstructs $|H_n(\xi)|^2 e^{-\xi^2}$ from laboratory data. And Hermite-Gaussians power "spectral" numerics: quantum chemists diagonalize anharmonic Hamiltonians in the $\psi_n$ basis — equation (7) doing production work — with Gauss–Hermite quadrature (nodes at the zeros of $H_n$) finishing the integrals to machine precision.

Think of the Gaussian $e^{-\xi^2/2}$ as the shape of the room the particle lives in: every oscillator bound state, however excited, dies off this same way at large distance, because out there the potential dominates and dictates the decay. The Hermite polynomial is the standing-wave pattern painted inside that room: $H_n$ has exactly $n$ sign changes, carving exactly $n$ nodes — the fingerprint of the $n$-th state. The polynomial wiggles; the Gaussian confines; the product is a bound state.

The division of labour is the point: the oscillator's family of "inside patterns" closes under differentiation and under multiplication by $\xi$. That closure is what the recurrence relations express, and it is why the oscillator is the one problem whose every matrix element can be written down without doing a single new integral.

Each state adds one node — more sign changes mean more curvature, and curvature is kinetic energy: that is why the states stack up. Parity alternates strictly: the potential is symmetric, so (as Parity Symmetry of One-Dimensional Eigenstates will formalize) every bound state is even or odd, and the family delivers exactly that: $H_n(-\xi) = (-1)^n H_n(\xi)$.

First, the natural coordinate. The oscillator owns one length, built from its only three constants; measuring position in units of it strips every dimension out of the mathematics:

where: $a = \sqrt{\hbar/m\omega}$ — the oscillator length, the scale on which the ground state is wide: picometres for a molecular vibration, tens of nanometres for an electron in a quantum dot. Every formula below lives in $\xi$; to return to the lab, multiply by $a$. The Quantum Harmonic Oscillator: Analytic Solution showed that writing $\psi = H(\xi)\,e^{-\xi^2/2}$ and demanding normalizability forces $H$ to be a polynomial — all we need to recall from there. The polynomial obeys the Hermite equation:

where: $n = 0, 1, 2, \dots$ — the degree of the polynomial and the number of the state. This second-order ODE has one polynomial solution per non-negative integer $n$ (the other solution grows like $e^{\xi^2}$ and is discarded by physics). Fix the overall scale by convention — leading coefficient $2^n$ — and the family is unique. You never solve equation (2) directly — the family has two compact "birth certificates". The first is the Rodrigues formula: differentiate a Gaussian $n$ times and dress the result:

where: the $(-1)^n$ keeps the leading coefficient positive. Each derivative of $e^{-\xi^2}$ pulls down $-2\xi$ or differentiates the prefactor, so the result is always (polynomial) × $e^{-\xi^2}$; the $e^{+\xi^2}$ strips the Gaussian off. Try $n=1$: $(-1)\,e^{\xi^2}(-2\xi e^{-\xi^2}) = 2\xi$. The second is the generating function — one exponential that carries the entire family in its Taylor expansion:

where: $t$ — a bookkeeping variable; $H_n$ is $n!$ times the coefficient of $t^n$. This is the most efficient line in the subject: differentiating it with respect to $t$ or $\xi$ produces every identity below in two lines each. Here is the family it produces:

Leading term always $(2\xi)^n$; parity always $(-1)^n$; always exactly $n$ real zeros, all inside the classical region.

Every physical question about the oscillator — is the state normalized, what is the probability of measuring $E_n$, how strongly does light drive the $n \to m$ transition — reduces to an integral of two Hermite-Gaussians against a power of $\xi$, and orthogonality answers them all at once:

where: $e^{-\xi^2}$ — the Gaussian weight, which is exactly the product of the two envelope factors $e^{-\xi^2/2}$ carried by $\psi_m$ and $\psi_n$; $\delta_{mn}$ — different states never overlap; $2^n n!\sqrt{\pi}$ — the diagonal value that fixes normalization constants. Note the weight: Hermite polynomials are not orthogonal on their own — only the full eigenfunctions, polynomial times Gaussian, are.

The payoff that turns spectroscopy into arithmetic — rearrange the three-term recurrence:

where: the left side is "the position operator acting on state $n$" in polynomial clothing. Multiplying a state by position produces only its two neighbours — nothing else. Sandwich equation (6) between eigenfunctions and orthogonality wipes out everything except $m = n \pm 1$:

where: $\sqrt{\hbar/2m\omega} = a/\sqrt{2}$ — the universal length of oscillator matrix elements. This is the selection rule $\Delta n = \pm 1$: an infrared photon, coupling $\propto \hat{x}$, moves a vibration one rung at a time — CO's single strong IR line at 2143 cm⁻¹ is equation (7). The same algebra is where Ladder Operators and the Algebraic Oscillator gets $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$: $\hat{a}, \hat{a}^\dagger \propto \xi \mp \mathrm{d}/\mathrm{d}\xi$ are the two recurrences in operator form. Two routes, one structure; neither re-derives the other. Iterating once more gives $\langle n|\hat{x}^2|n\rangle = (n+\tfrac12)\hbar/m\omega$ — how Zero-Point Energy reads the ground state's irreducible spread ($\langle x^2\rangle_0 = \hbar/2m\omega$) straight off this family.

That last row deserves its picture:

A classical oscillator spends most time near the turning points, where it moves slowest — the dashed curve's edge peaks. The quantum density oscillates, but its local average hugs the classical curve ever more closely as $n$ grows. The correspondence principle caught in the act — the same limit Infinite Square Well exhibited with flat-topped sines.

Example 1 — build, check and locate: the n = 3 state of HCl (confidence builder). The HCl vibration has wavenumber $\tilde{\nu} \approx 2990\,\mathrm{cm}^{-1}$ and reduced mass $\mu = 0.980\,\mathrm{u}$. Construct $H_3$ from the recurrence, and find where the nodes of $\psi_3$ sit in picometres.

Step 1. Build the polynomial with the three-term recurrence, no derivatives needed. From $H_0 = 1$, $H_1 = 2\xi$: $H_2 = 2\xi(2\xi) - 2(1)(1) = 4\xi^2 - 2$, then $H_3 = 2\xi(4\xi^2 - 2) - 2(2)(2\xi) = 8\xi^3 - 12\xi$.

Step 2. Nodes: $8\xi^3 - 12\xi = 4\xi(2\xi^2 - 3) = 0$ gives $\xi = 0$ and $\xi = \pm\sqrt{3/2} = \pm 1.225$.

Step 3. The length unit: $\omega = 2\pi c\tilde{\nu} = 5.63\times10^{14}\,\mathrm{rad/s}$, so $a = \sqrt{\hbar/\mu\omega} = \sqrt{1.055\times10^{-34}/(1.627\times10^{-27}\cdot 5.63\times10^{14})} = 10.7\,\mathrm{pm}$.

Step 4. Physical positions: $x = \xi a$, so the nodes sit at $x = 0$ and $\boxed{\pm 13.1\,\mathrm{pm}}$ — a few percent of the 127 pm bond length. For scale, the $n=2$ turning point is $\sqrt{5}\,a = 24.0\,\mathrm{pm}$: all nodes comfortably inside the classical region, as they must be.

Example 2 — how much does a "rigid" bond actually move? (exam level). The CO molecule has $\tilde{\nu} = 2143\,\mathrm{cm}^{-1}$, reduced mass $\mu = 6.86\,\mathrm{u}$, and bond length $113\,\mathrm{pm}$. Find the RMS displacement of the bond in its vibrational ground state, and the probability that a measurement finds the bond stretched beyond its classical limit.

Step 1. Frequency: $\omega = 2\pi c\tilde{\nu} = 2\pi (2.998\times10^{10}\,\mathrm{cm/s})(2143\,\mathrm{cm}^{-1}) = 4.04\times10^{14}\,\mathrm{rad/s}$.

Step 2. The ground state is a pure Gaussian with $\langle x^2\rangle = \hbar/2\mu\omega$ (equation 7 applied twice): $\langle x^2\rangle = 1.055\times10^{-34}/(2 \cdot 1.139\times10^{-26} \cdot 4.04\times10^{14}) = 1.147\times10^{-23}\,\mathrm{m}^2$.

Step 3. RMS displacement: $x_{\mathrm{rms}} = \sqrt{\langle x^2\rangle} = \boxed{3.4\,\mathrm{pm}}$ — the "rigid" CO bond breathes by about 3.0% of its length even at absolute zero. This zero-point motion is directly visible as the Debye–Waller attenuation in X-ray and neutron scattering.

Step 4. Beyond the turning point: for $n=0$ the classical amplitude is $\xi = 1$. The leak-out probability is $P = 1 - \mathrm{erf}(1) = \boxed{0.157}$ — nearly 16% of measurements catch the bond in classically forbidden stretch, pure Gaussian-tail physics with no polynomial in sight ($H_0 = 1$).

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §2.3 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.5 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 7 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. V · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 7 · Abramowitz & Stegun, Handbook of Mathematical Functions, Ch. 22

Provenance. Equations and conventions cross-checked against Griffiths 3rd ed. §2.3 (eigenfunctions, normalization Eq. 2.85), Shankar 2nd ed. Ch. 7 (series solution, recursion $a_{k+2}/a_k$), Sakurai 2nd ed. §2.5 (ladder operators), and Abramowitz & Stegun Ch. 22 (Rodrigues 22.11, generating function 22.9, recurrence 22.7, orthogonality 22.2.14). A verifier should recompute: (1) the recursion $a_{k+2}=(2k+1-\varepsilon)a_k/[(k+1)(k+2)]$ and its $2/k$ asymptotics; (2) the Rodrigues integration-by-parts giving $\sqrt{\pi}2^n n!$; (3) $H_0=1,H_1=2\xi,H_2=4\xi^2-2,H_3=8\xi^3-12\xi$ from the recurrence; (4) $\langle n+1|\hat x|n\rangle=\sqrt{(n+1)\hbar/2m\omega}$; (5) classical turning point $A=\sqrt{(2n+1)\hbar/m\omega}$ from $E_n=\tfrac12 m\omega^2 A^2$. All five, plus every worked number, node position and orthogonality integral on this page, were re-verified symbolically (sympy) and numerically for this edition.

Builds on: Quantum Harmonic Oscillator: Analytic Solution · Time-Independent Schrödinger Equation in One Dimension · Bound States versus Scattering States

Leads to: Ladder Operators and the Algebraic Oscillator · Zero-Point Energy · Wave Packets and Their Construction · Coherent States of the Harmonic Oscillator