Hermite Polynomials

Definition

The Hermite polynomials H_n(x), n = 0, 1, 2, …, are polynomials of degree n on ℝ, orthogonal with respect to the Gaussian weight w(x) = e^{−x²}:

∫_{−∞}^{∞} H_n(x) H_m(x) e^{−x²} dx = √π · 2ⁿ · n! · δ_{nm}

Three equivalent ways to define them (physicists' convention):

1. Gram–Schmidt on 1, x, x², … with the inner product ⟨f, g⟩ = ∫ f g e^{−x²} dx.
2. Rodrigues formula: H_n(x) = (−1)ⁿ e^{x²} dⁿ/dxⁿ [e^{−x²}].
3. Hermite ODE: bounded solutions of H'' − 2x H' + 2n H = 0 on ℝ.

Two conventions exist:

• Physicists' Hermite H_n. Weight e^{−x²}; leading coefficient 2ⁿ. Used here.
• Probabilists' Hermite He_n. Weight e^{−x²/2}; leading coefficient 1. He_n(x) = 2^{−n/2} H_n(x/√2).

Check which one a textbook, paper, or library uses before computing.

The first few polynomials

H₀(x) = 1
H₁(x) = 2x
H₂(x) = 4x² − 2
H₃(x) = 8x³ − 12x
H₄(x) = 16x⁴ − 48x² + 12
H₅(x) = 32x⁵ − 160x³ + 120x

The recurrence

H_{n+1}(x) = 2x H_n(x) − 2n H_{n−1}(x)

generates them all from H₀ = 1 and H₁ = 2x. Try n = 1: H₂ = 2x(2x) − 2(1)(1) = 4x² − 2. ✓ H_n alternates parity: even for even n, odd for odd n. Each H_n has exactly n distinct real roots.

Rodrigues formula

H_n(x) = (−1)ⁿ e^{x²} · dⁿ/dxⁿ [e^{−x²}]

Check n = 2: d/dx [e^{−x²}] = −2x e^{−x²}. d²/dx² [e^{−x²}] = (4x² − 2) e^{−x²}. Multiply by (−1)² e^{x²} = e^{x²}: H₂ = 4x² − 2. ✓

The pattern — apply n derivatives to the weight function, multiply by the inverse weight — generates orthogonal polynomial families across the board (Legendre with (x² − 1)ⁿ, Laguerre with xⁿ e^{−x}). The same Rodrigues-style move drops out of the Sturm–Liouville framework when the weight satisfies a particular ODE.

Hermite differential equation

H_n is the unique polynomial solution to

H''(x) − 2x H'(x) + 2n H(x) = 0

This isn't quite Sturm–Liouville form. Multiply through by e^{−x²}:

(e^{−x²} H'(x))' + 2n e^{−x²} H(x) = 0

Now it's Sturm–Liouville with p(x) = e^{−x²}, weight ρ(x) = e^{−x²}, eigenvalue 2n. The weight ensuring orthogonality on the unbounded line ℝ is exactly the Gaussian — without it, the polynomials would not have finite norms.

Quantum harmonic oscillator

The 1D harmonic-oscillator Hamiltonian is H = p̂²/(2m) + ½ m ω² x̂². The time-independent Schrödinger equation is

−(ℏ²/(2m)) ψ''(x) + ½ m ω² x² ψ(x) = E ψ(x)

Introduce dimensionless variable u = x √(mω/ℏ) and dimensionless energy ε = 2E/(ℏω):

ψ''(u) + (ε − u²) ψ(u) = 0

Asymptotic analysis suggests ψ(u) → e^{−u²/2} times a polynomial. Substitute ψ(u) = H(u) e^{−u²/2}, and the equation for H becomes

H''(u) − 2u H'(u) + (ε − 1) H(u) = 0

— Hermite's equation with 2n = ε − 1, so ε = 2n + 1. Translating back:

E_n = ℏω (n + 1/2),     n = 0, 1, 2, …
ψ_n(x) = (1/√(2ⁿ n! √π)) · H_n(x) · e^{−x²/2}

(with x in dimensionless units). The QHO has discrete energy levels equally spaced by ℏω, starting at the zero-point energy ½ℏω. The wavefunctions are exactly Hermite polynomials times a Gaussian envelope.

|ψ_n(x)|² — quantum vs classical

Plot |ψ_n|² for n = 0, 1, 2, 3, 4:

• n = 0: A single Gaussian peak at x = 0.
• n = 1: Two peaks, one node at x = 0.
• n = 2: Three peaks, two nodes — H_2 has roots at ±1/√2.
• n = 3: Four peaks, three nodes.
• n = 4: Five peaks, four nodes.

Compare with the classical-oscillator probability density: a particle of energy E spends time ∝ 1/v(x) at position x, peaked at the turning points ±x_max. The quantum ground state is the opposite — peaked in the middle, where the classical oscillator moves fastest.

As n grows, the quantum density develops more oscillations, and its envelope starts to match the classical 1/v shape — Bohr's correspondence principle. By n ≈ 50, the envelope tracks the classical distribution almost perfectly.

Ladder operators

Define the dimensionless lowering and raising operators

a  = (x + d/dx)/√2        (lowering)
a† = (x − d/dx)/√2        (raising)

[a, a†] = 1

The Hamiltonian is H = ℏω (a† a + 1/2). The ground state ψ₀(x) = π^{−1/4} e^{−x²/2} satisfies a ψ₀ = 0 — the operator "ladder" terminates downward at ψ₀. Higher states are built by repeated application of a†:

ψ_n(x) = (a†)ⁿ ψ₀(x) / √(n!)

Each application of a† brings the next Hermite polynomial into existence. Operator algebra and Hermite polynomials are two sides of the same coin — the operator view is what generalizes cleanly to many-particle states and to quantum field theory.

Gauss–Hermite quadrature

For integrals of the form ∫_{−∞}^{∞} f(x) e^{−x²} dx, the optimal n-point rule is

∫_{−∞}^{∞} f(x) e^{−x²} dx ≈ ∑_{i=1}^{n} w_i f(x_i)

where the x_i are the n roots of H_n(x) and

w_i = 2^{n−1} n! √π / (n² H_{n−1}(x_i)²)

The rule is exact for any polynomial f of degree ≤ 2n − 1. To integrate ∫ g(x) e^{−x²/(2σ²)} dx (a non-unit Gaussian), substitute and re-scale to the canonical form. For Bayesian model evidence ∫ p(D|θ) p(θ) dθ where p(θ) is approximately Gaussian, the Laplace approximation plus Gauss–Hermite is a textbook quadrature technique.

Worked example — compute ∫ x² e^{−x²} dx

Two ways:

Exact. ∫_{−∞}^{∞} x² e^{−x²} dx = (√π) / 2 ≈ 0.886227.

Gauss-Hermite (n = 2). Roots of H₂(x) = 4x² − 2 are x = ±1/√2. Weights: w_1 = w_2 = √π / 2. Then

∑ w_i x_i² = (√π/2)(1/2) + (√π/2)(1/2) = √π/2  ≈ 0.886227  ✓

Exact, because the integrand is f(x) e^{−x²} with f(x) = x² — a polynomial of degree 2 ≤ 2n − 1 = 3 for n = 2. For more complicated f, increase n; the error decays super-algebraically for smooth integrands that don't grow too fast.

Hermite vs related orthogonal polynomials

	Hermite H_n	Legendre P_n	Laguerre L_n	Chebyshev T_n
Interval	(−∞, ∞)	[−1, 1]	[0, ∞)	[−1, 1]
Weight w(x)	e^{−x²}	1	e^{−x}	1/√(1−x²)
Leading coeff	2ⁿ	(2n)!/(2ⁿ n!²)	(−1)ⁿ/n!	2^{n−1}
Quantum role	QHO eigenstates	Spherical harmonics	Hydrogen radial	(not standard)
Quadrature	Gauss-Hermite	Gauss-Legendre	Gauss-Laguerre	Clenshaw–Curtis
Rodrigues	(−1)ⁿ e^{x²} (e^{−x²})⁽ⁿ⁾	(2ⁿ n!)⁻¹ ((x²−1)ⁿ)⁽ⁿ⁾	(e^x / n!) (xⁿ e^{−x})⁽ⁿ⁾	cos(n arccos x)
Common application	QHO; Bayesian quadrature	Multipole expansion	Atomic radials	Polynomial fitting

Where Hermite polynomials appear

• Quantum harmonic oscillator. ψ_n(x) ∝ H_n(x) e^{−x²/2}, E_n = ℏω(n + 1/2). The single most-studied bound-state system in quantum mechanics; building block of phonons, photons, and quantum-field-theory mode expansions.
• Hermite functions ψ_n in signal processing. ψ_n form an orthonormal basis of L²(ℝ) — the eigenfunctions of the continuous Fourier transform! Used in time-frequency analysis (Wigner distributions, Gabor expansions).
• Gauss–Hermite quadrature. Optimal integration against a Gaussian weight. Used in Bayesian inference, kernel methods, statistical physics.
• Probability and combinatorics. Probabilists' He_n are the orthogonal polynomials for the standard normal distribution — appearing in Edgeworth and Gram–Charlier series for distribution approximation.
• Mehler kernel. The Schrödinger propagator of the QHO has a closed-form Hermite-polynomial expansion (Mehler's formula, 1866) — a rare exactly-solvable evolution operator.
• Stochastic calculus. Probabilists' Hermite polynomials in Brownian motion appear in the Wiener chaos decomposition and Malliavin calculus.
• Quantum optics. Photon-number states |n⟩ — eigenstates of the harmonic-oscillator quantization of a single EM mode — have Hermite-polynomial position-space wavefunctions.

Common mistakes

• Mixing physicists' and probabilists' conventions. H_n with weight e^{−x²} (physicists, used here) vs He_n with weight e^{−x²/2} (probabilists). Coefficients and norms differ by powers of √2 and 2. Wolfram, NIST, NumPy use the physicists' form; statistics references often use probabilists'.
• Forgetting the Gaussian envelope on QHO wavefunctions. H_n(x) grows without bound for large x — only the e^{−x²/2} prefactor makes ψ_n normalizable. Just H_n by itself is not the wavefunction.
• Computing H_n via Rodrigues for large n. Symbolic differentiation of e^{−x²} works for small n but is slow. Use the recurrence H_{n+1} = 2x H_n − 2n H_{n−1} for n > 10 or so.
• Sloppy normalization in ψ_n. The standard physicists' normalization is ψ_n(x) = (1/√(2ⁿ n! √π)) H_n(x) e^{−x²/2}. The 2ⁿ n! √π factor matters whenever you compute expectation values.
• Using Gauss–Hermite for non-Gaussian-decaying integrands. Gauss–Hermite efficiency assumes the integrand is bounded times a Gaussian. If your integrand decays slower than e^{−x²}, the quadrature loses accuracy; if it decays faster, you're computing unnecessarily — sometimes a simpler trapezoidal rule on a truncated interval is just as good.