Hermite Polynomials

Hermite polynomials

H_{n} (x)

are a family of orthogonal polynomials arising from the weight

e^{- x^{2}}

on the real line. They are eigenfunctions of the differential operator behind the quantum harmonic oscillator and the Gaussian, and form a complete orthogonal basis for

L^{2} (R, e^{- x^{2}})

. Used in quantum mechanics, probability, numerical quadrature, and signal processing. Named after Charles Hermite (1864), though Laplace and Chebyshev anticipated them.

The Hermite polynomials are the orthogonal polynomial sequence associated with the Gaussian weight function. There are two standard conventions: the physicists' $H_{n} (x)$ (weight $e^{- x^{2}}$ ) and the probabilists' $H e_{n} (x)$ (weight $e^{- x^{2} /2}$ ). They satisfy a second-order linear ODE, a three-term recurrence, a Rodrigues formula, and an explicit generating function. Their deep importance comes from the fact that $H_{n} (x) e^{- x^{2} /2}$ are the stationary states of the quantum harmonic oscillator — turning a polynomial identity into a description of vibrating quantum systems. A surprising fact: the Hermite functions are eigenfunctions of the Fourier transform, with eigenvalues that are just the powers of $i$ .

The Core Insight

A single weight function — the Gaussian — determines, by demanding orthogonality, an entire infinite family of polynomials. Because the Gaussian is the unique fixed point of the Fourier transform and the ground state of the harmonic oscillator, these polynomials inherit extraordinary structure: they diagonalize differentiation-plus-multiplication operators, encode the algebra of raising and lowering, and are simultaneously analytic, algebraic, and physical objects.

Formal Statement

Physicists' Hermite Polynomials — Rodrigues Formula

The physicists' Hermite polynomials are defined by the Rodrigues formula

H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d ^{n}}{d x ^{n}} e^{- x^{2}}, n = 0, 1, 2, \dots

⟳each derivative of

e^{- x^{2}}

produces

e^{- x^{2}}

times a polynomial; the prefactor

(- 1)^{n} e^{x^{2}}

strips the exponential, leaving a degree-

n

polynomial with leading coefficient

2^{n}

Here $x \in R$ , $n$ is the degree (a non-negative integer), and the leading term of $H_{n}$ is $2^{n} x^{n}$ .

Explicit Sum and First Few Polynomials

The closed-form expansion is

H_{n} (x) = n! m = 0 \sum ⌊ n /2 ⌋ \frac{( - 1 ) ^{m}}{m ! ( n - 2 m )!} (2 x)^{n - 2 m} .

⟳substitute the Maclaurin series of the generating function

e^{2 x t - t^{2}}

and collect the coefficient of

t^{n} / n!

, using the Cauchy product of

e^{2 x t}

and

e^{- t^{2}}

The first six are:

H_{0} H_{3} = 1, = 8 x^{3} - 12 x, H_{1} H_{4} = 2 x, = 16 x^{4} - 48 x^{2} + 12, H_{2} H_{5} = 4 x^{2} - 2, = 32 x^{5} - 160 x^{3} + 120 x .

⟳apply the recurrence [6] starting from

H_{0} = 1, H_{1} = 2 x

, or expand [2] term by term.

Probabilists' Convention

The probabilists' Hermite polynomials use the standard-normal weight:

H e_{n} (x) = (- 1)^{n} e^{x^{2} /2} \frac{d ^{n}}{d x ^{n}} e^{- x^{2} /2} .

⟳identical to [1] with

e^{- x^{2} /2}

in place of

e^{- x^{2}}

; the scaling

x \mapsto x / 2

relates the two families.

The two conventions are related by

H_{n} (x) = 2^{n /2} H e_{n} (2 x), H e_{n} (x) = 2^{- n /2} H_{n} (x / 2) .

⟳substitute the change of variable

u = 2 x

into the Rodrigues formulas and compare derivative chains.

Defining Properties

The physicists' Hermite polynomials satisfy:

(a) Three-term recurrence:

H_{n + 1} (x) = 2 x H_{n} (x) - 2 n H_{n - 1} (x) .

⟳differentiate the generating function [9] with respect to

t

and equate coefficients of

t^{n}

(b) Differential equation (Hermite's ODE):

H_{n}^{''} - 2 x H_{n}^{'} + 2 n H_{n} = 0.

⟳substitute

y = H_{n}

into the Sturm–Liouville form

(e^{- x^{2}} y^{'})^{'} + 2 n e^{- x^{2}} y = 0

and expand the derivative.

(c) Derivative relation:

H_{n}^{'} (x) = 2 n H_{n - 1} (x) .

⟳differentiate the Rodrigues formula or the generating function with respect to

x

Generating Function and Orthogonality

Generating function:

e^{2 x t - t^{2}} = n = 0 \sum \infty H_{n} (x) \frac{t ^{n}}{n !} .

⟳write

e^{2 x t - t^{2}} = e^{x^{2}} e^{- (x - t)^{2}}

and Taylor-expand

e^{- (x - t)^{2}}

t

about

t = 0

; the

n

-th coefficient is

(- 1)^{n} e^{x^{2}} \partial_{x}^{n} e^{- x^{2}} / n!

Orthogonality:

\int_{- \infty}^{\infty} H_{m} (x) H_{n} (x) e^{- x^{2}} d x = 2^{n} n! π δ_{mn} .

⟳insert the Rodrigues form of one factor, integrate by parts

n

times (boundary terms vanish from the Gaussian decay), and use

\int e^{- x^{2}} H_{n}^{(n)}

telescoping to

\int e^{- x^{2}} (2^{n} n!) d x

Why It Holds / Derivation

Orthogonality from the Rodrigues Formula

We prove [10] for $m \leq n$ .

Substitute [1] for $H_{n}$ :

I = \int_{- \infty}^{\infty} H_{m} (x) (- 1)^{n} \frac{d ^{n}}{d x ^{n}} e^{- x^{2}} d x .

⟳the factor

e^{x^{2}}

in the Rodrigues formula cancels the weight

e^{- x^{2}}

Integrate by parts $n$ times. Each boundary term contains a polynomial times $e^{- x^{2}}$ , which vanishes at $\pm \infty$ :

I = (- 1)^{n} (- 1)^{n} \int_{- \infty}^{\infty} (\frac{d ^{n}}{d x ^{n}} H_{m} (x)) e^{- x^{2}} d x = \int_{- \infty}^{\infty} H_{m}^{(n)} (x) e^{- x^{2}} d x .

⟳each integration by parts transfers one derivative from

e^{- x^{2}}

onto

H_{m}

, contributing a sign

(- 1)

;

n

of them cancel the prefactor.

If $m < n$ , then $H_{m}^{(n)} \equiv 0$ since $de g H_{m} = m$ , so $I = 0$ .

If $m = n$ , then $H_{n}^{(n)} = 2^{n} n!$ (the $n$ -th derivative of $2^{n} x^{n}$ ), giving

I = 2^{n} n! \int_{- \infty}^{\infty} e^{- x^{2}} d x = 2^{n} n! π .

⟳

\int_{- \infty}^{\infty} e^{- x^{2}} d x = π

is the Gaussian integral.

By symmetry of the roles of $m, n$ this establishes [10]. $■$

The Differential Equation from the Recurrences

Combine [6] and [8].

From [8], $H_{n - 1} = H_{n}^{'} / (2 n)$ , so substituting into [6]:

H_{n + 1} = 2 x H_{n} - H_{n}^{'} .

⟳replace

2 n H_{n - 1}

in [6] using [8].

Differentiate [14] and use [8] with $n \to n + 1$ , $H_{n + 1}^{'} = 2 (n + 1) H_{n}$ :

2 (n + 1) H_{n} = 2 H_{n} + 2 x H_{n}^{'} - H_{n}^{''} .

⟳

\frac{d}{d x} (2 x H_{n}) = 2 H_{n} + 2 x H_{n}^{'}

Rearranging gives Hermite's ODE [7]:

H_{n}^{''} - 2 x H_{n}^{'} + 2 n H_{n} = 0. ■

⟳collect terms and cancel the common factor of

2

Charles Hermite | 1822–1901

A French mathematician who never earned a formal degree in the standard way (he struggled with examinations) yet became one of the great analysts of the 19th century. In 1864 he studied these polynomials in connection with continued fractions and approximation, though Laplace had encountered them in 1810 in probability and Chebyshev in 1859. Hermite's broader legacy includes the first proof that $e$ is transcendental (1873) and foundational work on elliptic functions and the general quintic. The polynomials, the Hermite normal form, Hermitian matrices, and Hermite interpolation all bear his name. His insistence on the centrality of "natural" analytic objects — the Gaussian above all — shaped the subject.

Building Intuition

Think of orthogonal polynomials as a Gram–Schmidt process applied to the monomials $1, x, x^{2}, \dots$ but with an inner product weighted by the Gaussian. The weight $e^{- x^{2}}$ suppresses large $x$ , so the "geometry" of function space is curved toward the origin. The Hermite polynomials are simply the orthonormalized monomials in this geometry — $H_{0}$ is the constant, $H_{1}$ is the linear function orthogonal to it, $H_{2}$ the quadratic orthogonal to both, and so on.

A second picture: the operator $\overset{a}{^}^{†} = x - \frac{1}{2} \frac{d}{d x}$ (a "raising operator") manufactures $H_{n + 1}$ from $H_{n}$ . Applied repeatedly to the Gaussian ground state, it generates the entire family. Each application adds one node (zero crossing), so $H_{n}$ has exactly $n$ real roots, all interlacing those of $H_{n + 1}$ — like the overtones of a vibrating string, each higher mode wiggling one more time.

A third picture: the parity is rigid. $H_{n} (- x) = (- 1)^{n} H_{n} (x)$ , so even-index polynomials are even functions and odd-index ones are odd. This mirrors the symmetric/antisymmetric standing waves of the harmonic oscillator.

:::svg[The first four physicists' Hermite polynomials $H_{0}$ through $H_{3}$ on $[- 2, 2]$ , showing increasing oscillation and parity alternation.]

Worked Examples

Computing an Integral via Orthogonality

Given: Evaluate

J = \int_{- \infty}^{\infty} x^{2} H_{2} (x) e^{- x^{2}} d x

. Step 1 — Express $x^{2}$ in the Hermite basis. From [3],

H_{2} = 4 x^{2} - 2

, so

x^{2} = \frac{1}{4} H_{2} + \frac{1}{2} H_{0} .

⟳solve

H_{2} = 4 x^{2} - 2

for

x^{2}

Step 2 — Substitute and use orthogonality [10].

J = \int_{- \infty}^{\infty} (\frac{1}{4} H_{2} + \frac{1}{2} H_{0}) H_{2} e^{- x^{2}} d x = \frac{1}{4} ∥ H_{2} ∥^{2} + \frac{1}{2} \cdot 0.

⟳the

H_{0} H_{2}

cross term vanishes since

0 \neq = 2

Step 3 — Plug in the norm $∥ H_{2} ∥^{2} = 2^{2} 2! π = 8 π$ .

J = \frac{1}{4} \cdot 8 π = 2 π .

⟳use [10] with

n = 2

J = 2 π \approx 3.5449

Quantum Harmonic Oscillator Ground and First Excited States

Given: The normalized eigenfunctions of the quantum harmonic oscillator (in units

ℏ = m = ω = 1

) are

ψ_{n} (x) = \frac{1}{2 ^{n} n ! π} H_{n} (x) e^{- x^{2} /2} .

⟳

ψ_{n}

solves

- \frac{1}{2} ψ^{''} + \frac{1}{2} x^{2} ψ = (n + \frac{1}{2}) ψ

when

H_{n}

solves [7]; the prefactor enforces

\int ∣ ψ_{n} ∣^{2} = 1

via [10].

Find $ψ_{0}$ , $ψ_{1}$ , and verify orthonormality of $ψ_{0}$ .

Step 1 — Ground state ( $n = 0$ ):

H_{0} = 1

, norm

1 \cdot 1 \cdot π = π^{1/4}

ψ_{0} (x) = π^{- 1/4} e^{- x^{2} /2} .

⟳

2^{0} 0! π = π

, take the square root.

Step 2 — First excited state ( $n = 1$ ):

H_{1} = 2 x

, norm

2 \cdot 1 \cdot π

ψ_{1} (x) = \frac{2 x}{2 π} e^{- x^{2} /2} = (\frac{4}{π})^{1/4} x e^{- x^{2} /2} .

⟳simplify

2/ 2 π = 2 / π^{1/4} = (4/ π)^{1/4}

Step 3 — Verify $\int ∣ ψ_{0} ∣^{2} = 1$ :

\int_{- \infty}^{\infty} π^{- 1/2} e^{- x^{2}} d x = π^{- 1/2} π = 1.

⟳Gaussian integral.

ψ_{0} = π^{- 1/4} e^{- x^{2} /2}, ψ_{1} = (4/ π)^{1/4} x e^{- x^{2} /2}

Energies: $E_{0} = \frac{1}{2}$ , $E_{1} = \frac{3}{2}$ (in units of $ℏ ω$ ).

Common Misconceptions

The two conventions are NOT interchangeable. Physicists'

H_{n}

use weight

e^{- x^{2}}

and have leading coefficient

2^{n}

; probabilists'

H e_{n}

use weight

e^{- x^{2} /2}

and leading coefficient

1

. Always check which one a textbook or software uses (e.g., NumPy numpy.polynomial.hermite is physicists', hermite_e is probabilists'). A factor-of-

2

scaling error from [5] silently corrupts every numerical result. The Rodrigues formula gives polynomials, not Gaussians. Students sometimes think

H_{n} (x)

"contains"

e^{- x^{2}}

. It does not — the exponentials cancel exactly.

H_{n}

is a pure polynomial; the Gaussian only appears in the weight and in the harmonic-oscillator wavefunctions

ψ_{n} = c_{n} H_{n} e^{- x^{2} /2}

. Orthogonality fails without the weight. One might guess

\int_{- \infty}^{\infty} H_{m} H_{n} d x = 0

for

m \neq = n

. This is false — indeed the integral diverges, since

H_{m} H_{n}

is a polynomial that grows without bound. For example

\int_{- \infty}^{\infty} H_{0} (x) H_{2} (x) d x = \int_{- \infty}^{\infty} (4 x^{2} - 2) d x = + \infty.

⟳a non-constant polynomial cannot have a finite integral over

R

; the Gaussian weight

e^{- x^{2}}

is essential for convergence and orthogonality.

Connections

Hermite polynomials sit inside the larger theory of Orthogonal Polynomials, alongside Legendre Polynomials (weight $1$ on $[- 1, 1]$ ), Laguerre Polynomials (weight $e^{- x}$ on $[0, \infty)$ ), and Chebyshev Polynomials. All arise from the Sturm-Liouville Theory framework as eigenfunctions of self-adjoint operators. Their orthogonality [10] is a special case of the general Gram-Schmidt Process applied with a weighted Inner Product Space.

In physics they are the bridge to the Quantum Harmonic Oscillator and, through the ladder operators, to the Creation and Annihilation Operators algebra. The Hermite functions $ψ_{n}$ are eigenfunctions of the Fourier Transform with eigenvalue $(- i)^{n}$ , linking these polynomials to harmonic analysis. In probability they generate the Edgeworth Expansion and the Wiener chaos decomposition of Gaussian Processes.

Real-World Applications

Quantum harmonic oscillator — $ψ_{n} = c_{n} H_{n} (x) e^{- x^{2} /2}$ are the exact stationary states; the foundation of quantum field theory's mode expansion and of vibrational spectroscopy.
Gaussian (Hermite) quadrature — the $n$ roots of $H_{n}$ are the optimal nodes for integrating $\int f (x) e^{- x^{2}} d x$ , exact for polynomials up to degree $2 n - 1$ ; used in computational physics and finance.
Image processing & optics — Hermite–Gaussian beams (TEM $_{mn}$ modes) describe laser transverse modes; Hermite functions form steerable filters in computer vision.
Probability & statistics — the Edgeworth/Gram–Charlier series expands non-Gaussian densities in $H e_{n}$ ; central to higher-order asymptotics.
Signal processing — the fractional Fourier transform is diagonalized by Hermite functions, enabling time–frequency analysis.
Stochastic calculus — $H e_{n} (B_{t} / t) t^{n /2}$ are martingales for Brownian motion $B_{t}$ ; the Wiener–Itô chaos expansion uses Hermite polynomials as a basis.
Molecular spectroscopy — Franck–Condon factors (overlaps of oscillator wavefunctions) are computed directly from Hermite polynomial integrals.

Historical Development

1810 — Pierre-Simon Laplace encounters the polynomials implicitly while studying probability and the normal distribution in Théorie analytique des probabilités.
1859 — Pafnuty Chebyshev studies them in the context of orthogonal expansions and least-squares approximation.
1864 — Charles Hermite publishes a systematic treatment connecting them to continued fractions and multivariate generalizations; the name becomes attached to him.
1880s — Thomas Stieltjes develops the general theory of orthogonal polynomials and the moment problem, with Hermite's polynomials as a key example.
1907–1910 — The Gram–Charlier and Edgeworth series formalize density approximation using Hermite polynomials in statistics.
1926 — Erwin Schrödinger solves the quantum harmonic oscillator; the Hermite functions become the canonical example of quantized energy levels.
1929 — Norbert Wiener uses Hermite functions in his homogeneous chaos theory, the seed of modern stochastic analysis.
1939 — Gábor Szegő's monograph Orthogonal Polynomials gives the definitive classical synthesis.
1947 — Wiener–Itô chaos expansion (Itô) places Hermite polynomials at the center of Malliavin calculus and stochastic PDEs.

Subtleties & Limits

While the one-dimensional theory is complete, several deeper questions remain active. The precise asymptotic distribution of the zeros of $H_{n}$ (Plancherel–Rotach asymptotics) connects to random matrix theory: the largest eigenvalue of a Gaussian unitary ensemble matrix and the largest Hermite zero share the Tracy–Widom distribution, and sharp universality results in this direction are still being refined. In several variables, no unique canonical generalization exists — there are tensor-product Hermite polynomials, Hermite polynomials for general covariance matrices, and the theory of multivariate orthogonal polynomials on weighted domains remains incomplete. Computationally, evaluating $H_{n} (x)$ for large $n$ and large $x$ suffers severe overflow/cancellation, and numerically stable algorithms (e.g., via the recurrence in log-space or asymptotic expansions) are an ongoing engineering concern in high-dimensional quadrature and spectral methods.