Hermite Polynomials

Hermite polynomials are the classical orthogonal family that lives on the whole real line, tamed there by the Gaussian weight $e^{-x^2}$ . They are one of the three great Rodrigues families — Legendre on an interval, Laguerre on a half-line, Hermite on all of $\mathbb{R}$ — and among them they carry the cleanest algebra, because their weight is its own Fourier transform and its derivative is proportional to itself. That is why the same functions surface, apparently unbidden, in quantum mechanics, signal processing, probability, and random-matrix theory. The physical reason they are unavoidable is short. Separate the time-independent Schrödinger equation for a quadratic potential, peel off the asymptotic Gaussian decay every bound state must have, and what remains is exactly Hermite's equation $y''-2xy'+2ny=0$ . The solutions terminate into polynomials only for the quantized energies $E_n=\hbar\omega(n+\tfrac12)$ , so the discreteness of the oscillator spectrum is just the analytic demand that $H_n$ be a polynomial rather than a runaway power series. The oscillator is in turn the backbone of quantum field theory, molecular vibrations, phonons, and laser modes — so Hermite's little ODE quietly underwrites a great deal of physics. Two master formulas generate everything else. The generating function $e^{2xt-t^2}=\sum H_n(x)\,t^n/n!$ delivers the recurrences, the differential equation, and special values by differentiation; the Rodrigues formula $H_n(x)=(-1)^n e^{x^2}\partial_x^n e^{-x^2}$ delivers orthogonality and the norm by integration by parts. One hazard runs through the subject: two rival normalizations, physicists' $H_n$ and probabilists' $He_n$ , differ by a $\sqrt2$ rescaling — mix them and every norm is wrong.

Key takeaways

Hermite polynomials $H_n(x)$ are the orthogonal polynomials on $(-\infty,\infty)$ with Gaussian weight $w(x)=e^{-x^2}$ (physicists' convention), degree $n$ , leading coefficient $2^n$ , parity $(-1)^n$ .
They satisfy Hermite's equation $y''-2xy'+2ny=0$ and are the eigenfunctions of the quantum harmonic oscillator, with the polynomial condition forcing $E_n=\hbar\omega(n+\tfrac12)$ .
A Rodrigues formula $H_n(x)=(-1)^n e^{x^2}\dfrac{\mathrm{d}^n}{\mathrm{d}x^n}e^{-x^2}$ and a generating function $e^{2xt-t^2}$ encode every identity by differentiation and integration by parts.
They obey the three-term recurrence $H_{n+1}=2xH_n-2nH_{n-1}$ , the derivative relation $H_n'=2nH_{n-1}$ , and orthogonality $\int H_mH_n e^{-x^2}\,\mathrm{d}x=2^n n!\sqrt{\pi}\,\delta_{mn}$ .
The Hermite functions $\psi_n=H_n(x)e^{-x^2/2}/\sqrt{2^n n!\sqrt\pi}$ form an orthonormal basis of $L^2(\mathbb{R})$ and diagonalize the Fourier transform with eigenvalue $(-i)^n$ .
Two conventions coexist — physicists' $H_n$ (weight $e^{-x^2}$ , $H_2=4x^2-2$ ) and probabilists' $He_n$ (weight $e^{-x^2/2}$ , monic, $He_2=x^2-1$ ) — related by $H_n(x)=2^{n/2}He_n(\sqrt2\,x)$ .
Key idea: Strip the Gaussian — writing the oscillator eigenfunction as $\psi(x)=H(x)e^{-x^2/2}$ turns the Schrödinger equation into Hermite's equation, and demanding that $H$ be a polynomial rather than a divergent series is exactly energy quantization.

The core insight

Strip the Gaussian: writing the oscillator eigenfunction as $\psi(x)=H(x)e^{-x^2/2}$ converts the Schrödinger equation into Hermite's equation, and the requirement that $H$ be a polynomial — not a divergent power series — is precisely energy quantization.

Where this lives today

Hermite polynomials are working tools across modern physics and computation. In numerical analysis, Gauss–Hermite quadrature places its nodes at the zeros of $H_n$ and integrates any function against $e^{-x^2}$ exactly up to degree $2n-1$ — the workhorse for Franck–Condon overlaps and Gaussian-weighted expectation values in statistical mechanics. In photonics, the Hermite–Gaussian $\mathrm{TEM}_{mn}$ modes are the transverse eigenmodes of stable laser resonators, and the same polynomials index structured-light beams used in optical tweezers, super-resolution microscopy, and spatial-mode multiplexing in fibre. In molecular spectroscopy, $\psi_n$ are the vibrational wavefunctions whose overlaps set infrared and Raman line intensities. And in random-matrix theory, $H_n$ are precisely the orthogonal polynomials of the Gaussian Unitary Ensemble: the Plancherel–Rotach asymptotics of $\psi_n$ produce both Wigner's semicircle law in the bulk and the Tracy–Widom distribution at the spectral edge — statistics that recur in the zeros of the Riemann zeta function, in growing-interface fluctuations, and in the largest eigenvalues of huge data-covariance matrices. The little ODE that quantized the oscillator now sits at the heart of universality theorems.

The physical picture

Two pictures anchor the family. The first is algebraic: treat $e^{-x^2}$ as an inner product that suppresses large $|x|$ , and Gram–Schmidt the monomials $1,x,x^2,\dots$ in it. What drops out, up to scale, is exactly $H_0,H_1,H_2,\dots$ — each $H_n$ the unique degree- $n$ polynomial orthogonal to all lower degrees. Because the weight is even, the polynomials inherit strict parity $H_n(-x)=(-1)^nH_n(x)$ , and each $H_n$ has $n$ real simple zeros that interlace with those of $H_{n+1}$ . The second picture is physical, and the one that makes Hermite polynomials inescapable. Attach the shared envelope $e^{-x^2/2}$ to each polynomial and you get the Hermite functions $\psi_n$ — the stationary states of the quantum harmonic oscillator. The ground state $\psi_0$ is a bare Gaussian; each higher state gains one more node, and the whole tower is generated by a raising operator $a^\dagger=\tfrac{1}{\sqrt2}(x-\partial_x)$ acting on the one below. The first figure is that tower: parabolic well, equally spaced rungs, and the wavefunction riding each rung.

Figure

The quantum harmonic oscillator ladder — each Hermite function ψₙ = Hₙ(x)·e^(−x²/2) sits at energy Eₙ = ħω(n+½), the rungs are equally spaced by ħω, and the ground state E₀ = ½ħω is the nonzero zero-point energy; state ψₙ has exactly n nodes

The recurrence, the derivative relation, and the orthogonality of the states are not separate facts but one algebra — that of the ladder operators — seen several ways. The second figure strips the potential away and lays the first four Hermite functions flat on a common axis, so the structural fact is visible directly: $\psi_n$ crosses zero exactly $n$ times, the curves alternate even and odd, and all sit inside the same Gaussian envelope $\psi_0(x)\propto e^{-x^2/2}$ .

Figure

The first four Hermite functions ψ₀…ψ₃ on a common axis, all bound inside the shared Gaussian envelope ±ψ₀ (dashed): ψ₀ is even with 0 nodes, ψ₁ odd with 1 node, ψ₂ even with 2 nodes, ψ₃ odd with 3 nodes at x = 0 and x = ±√(3/2) ≈ ±1.225 — the n real simple zeros that become Gauss–Hermite quadrature points

The core equations

Everything begins with orthogonality: $H_n$ is defined, up to scale, by being orthogonal to all lower degrees against the Gaussian weight. Fixing the scale by the leading coefficient $2^n$ fixes the norm too — the constant every quantum normalization divides out.

\int_{-\infty}^{\infty} H_m(x)\,H_n(x)\,e^{-x^2}\,\mathrm{d}x = 2^{n}\,n!\,\sqrt{\pi}\;\delta_{mn} \qquad (1)

where: $H_n$ — physicists' Hermite polynomial of degree $n$ ; $e^{-x^2}$ — Gaussian weight on $\mathbb{R}$ ; $\delta_{mn}$ — Kronecker delta (1 if $m=n$ , else 0); $\sqrt{\pi}=\Gamma(\tfrac12)$ — the value of the Gaussian integral. The off-diagonal zeros are orthogonality; the diagonal $2^n n!\sqrt\pi$ is the squared norm, growing factorially with $n$ . The most compact way to produce the polynomials is Rodrigues' formula: differentiate the Gaussian $n$ times and divide it back out. It makes degree, parity, and leading coefficient manifest at once.

H_n(x) = (-1)^n\,e^{x^2}\,\frac{\mathrm{d}^n}{\mathrm{d}x^n}\,e^{-x^2} \qquad (2)

where: $\frac{\mathrm{d}^n}{\mathrm{d}x^n}$ — the $n$ -th derivative in $x$ ; $(-1)^n$ — the sign making the leading coefficient $+2^n$ . Each derivative pulls down $\approx-2x$ , so the top term is $(-1)^n(-2x)^n=2^n x^n$ — degree $n$ , parity $(-1)^n$ built in. The generating function packages the whole sequence into one analytic object. Expanding $e^{2xt-t^2}$ in powers of $t$ reads off $H_n$ as the coefficient of $t^n/n!$ , and differentiating in $x$ or $t$ mints every identity below.

e^{2xt-t^2} = \sum_{n=0}^{\infty} H_n(x)\,\frac{t^n}{n!} \qquad (3)

where: $t$ — an auxiliary (formal) expansion variable; $x$ — the argument. Setting $x=0$ gives $e^{-t^2}=\sum H_n(0)t^n/n!$ , from which $H_{2m}(0)=(-1)^m(2m)!/m!$ falls out at once. Differentiating (3) in $t$ ties three consecutive degrees together — the three-term recurrence every classical orthogonal family satisfies, and the fastest way to tabulate the polynomials.

H_{n+1}(x) = 2x\,H_n(x) - 2n\,H_{n-1}(x) \qquad (4)

where: $n\ge1$ — the degree index. Seeded with $H_0=1$ , $H_1=2x$ , it marches upward: $H_2=4x^2-2$ , $H_3=8x^3-12x$ , $H_4=16x^4-48x^2+12$ . Differentiating (3) in $x$ gives the derivative relation — the analytic shadow of the lowering operator, since it drops the degree by one.

H_n'(x) = 2n\,H_{n-1}(x) \qquad (5)

where: $H_n'=\mathrm{d}H_n/\mathrm{d}x$ . Combined with (4), this closes the algebra: two first-order relations that imply the second-order ODE below. Eliminating the neighbours between (4) and (5) leaves the single second-order equation each $H_n$ obeys — Hermite's differential equation, with the degree appearing as the eigenvalue.

y'' - 2x\,y' + 2n\,y = 0 \qquad (6)

where: $y=H_n(x)$ — the polynomial solution; $2n$ — the eigenvalue selected by demanding a terminating solution. Non-integer eigenvalues give the second solution, which grows like $e^{x^2}$ and is discarded as non-normalizable. Attaching the shared envelope and dividing by the norm turns the polynomials into an orthonormal basis of square-integrable functions — the oscillator eigenstates.

\psi_n(x)=\frac{1}{\sqrt{2^n\,n!\,\sqrt{\pi}}}\,H_n(x)\,e^{-x^2/2} \qquad (7)

where: $\psi_n$ — the normalized Hermite function, $\langle\psi_m,\psi_n\rangle=\delta_{mn}$ in $L^2(\mathbb{R})$ ; $e^{-x^2/2}$ — the shared Gaussian envelope. The prefactor is exactly $1/\|H_n\|$ from (1). Finally, these same functions are eigenfunctions of the Fourier transform — the deepest reason the Gaussian ( $n=0$ ) is its fixed point.

\mathcal{F}[\psi_n](k)=(-i)^n\,\psi_n(k) \qquad (8)

where: $\mathcal{F}$ — the unitary (symmetric-convention) Fourier transform; $(-i)^n$ — the eigenvalue, periodic in $n$ mod 4, cycling $1,-i,-1,i$ . The self-similarity of $e^{-x^2/2}$ under $\mathcal{F}$ propagates up the whole ladder.

Derivation AFrom the oscillator to Hermite's equation and quantization

The most consequential fact in this article is that Hermite polynomials are what survives of the quantum harmonic oscillator once its unavoidable Gaussian tail is divided out — and that the innocuous requirement "the leftover must be a polynomial" is identical to energy being quantized. The goal is to start from the dimensionless time-independent Schrödinger equation for a quadratic well, insert the ansatz $\psi=H(x)e^{-x^2/2}$ , watch the potential cancel exactly against a piece of the second derivative to expose Hermite's equation, and then show by a power-series argument that a normalizable solution exists only for eigenvalue $2n$ , giving $E_n=\hbar\omega(n+\tfrac12)$ .

Step 1. Choose units $m=\hbar=\omega=1$ so the oscillator reads $-\tfrac12\psi''+\tfrac12 x^2\psi=\tfrac{\varepsilon}{2}\psi$ , where $\varepsilon=2E/\hbar\omega$ is the dimensionless energy. Why: nondimensionalizing removes every physical constant so the mathematics is bare — all that remains to be determined is the pure number $\varepsilon$ .

Step 2. Examine large $|x|$ : there the $\tfrac12 x^2\psi$ term dominates and $\psi''\approx x^2\psi$ , whose decaying solution is $\psi\sim e^{-x^2/2}$ . This forces the ansatz $\psi=H(x)e^{-x^2/2}$ with $H$ growing slower than the Gaussian decays. Why: every bound state must vanish at infinity, and the Gaussian is the only decaying asymptotic the equation allows — so factoring it out is not a trick but the unique way to isolate the non-trivial part.

Step 3. Differentiate the ansatz twice: $\psi''=(H''-2xH'+(x^2-1)H)e^{-x^2/2}$ . Substitute into Step 1 and note the $x^2H$ piece cancels precisely against the potential's $x^2\psi$ , leaving $H''-2xH'+(\varepsilon-1)H=0$ . Why: this exact cancellation of the $x^2$ terms is the whole point — it converts a problem with an unbounded potential into Hermite's equation, a polynomial eigenvalue problem with parameter $\varepsilon-1$ .

Step 4. Try a Frobenius series $H=\sum_k a_k x^k$ ; matching powers gives the two-term recursion $a_{k+2}=\dfrac{2k-(\varepsilon-1)}{(k+1)(k+2)}\,a_k$ . For large $k$ the ratio $a_{k+2}/a_k\to 2/k$ , which is exactly the coefficient growth of $e^{x^2}$ . Why: an unterminated series therefore behaves like $e^{x^2}$ , so $\psi=He^{-x^2/2}$ would blow up like $e^{+x^2/2}$ — non-normalizable and unphysical.

Step 5. The only escape is termination: the numerator must vanish at some $k=n$ , forcing $\varepsilon-1=2n$ , hence $E_n=\hbar\omega(n+\tfrac12)$ ; the surviving polynomial is $H_n$ . Why: discreteness is not imposed — it is the sole condition under which a square-integrable solution exists, so the quantized ladder and the polynomial nature of $H_n$ are the same statement. *As a consistency check: the recursion ratio $a_{k+2}/a_k\to 2/k$ matches the Taylor coefficients of $e^{x^2}$ , confirming that an unterminated series diverges like $e^{x^2}$ ; termination at degree $n$ then reproduces $E_n=\hbar\omega(n+\tfrac12)$ .*

Go deeper · Derivation B · 1 minOrthogonality and the norm from Rodrigues and the Gamma function

Derivation BOrthogonality and the norm from Rodrigues and the Gamma function

The aim is to obtain both halves of equation (1) — the vanishing cross integrals and the exact norm $2^n n!\sqrt\pi$ — from one tool, the Rodrigues formula fed through repeated integration by parts, with the Gamma function entering through $\sqrt\pi=\Gamma(\tfrac12)$ . The strategy: replace one Hermite factor by its Rodrigues form, integrate by parts $n$ times so the derivatives land on the other polynomial factor, and read off the answer from how many derivatives that polynomial can survive.

Step 1. Take $m\le n$ and write $\int H_mH_n e^{-x^2}\,\mathrm{d}x=(-1)^n\int H_m\,\partial_x^n e^{-x^2}\,\mathrm{d}x$ using Rodrigues for the single factor $H_n$ . Why: trading the polynomial $H_n$ for $n$ derivatives of the Gaussian is what lets integration by parts move all the differentiation onto $H_m$ , whose degree we control.

Step 2. Integrate by parts $n$ times; every boundary term carries a factor $e^{-x^2}$ times a polynomial and so vanishes at $\pm\infty$ , leaving $\int(\partial_x^n H_m)\,e^{-x^2}\,\mathrm{d}x$ . Why: the Gaussian's decay kills all boundary contributions cleanly — this is exactly why the whole real line, weighted by $e^{-x^2}$ , behaves like a compact interval for these integrals.

Step 3. If $m<n$ then $\partial_x^n H_m=0$ , because differentiating a degree- $m$ polynomial $n>m$ times annihilates it — so the integral vanishes and orthogonality is proved. Why: this is the structural heart of orthogonality — a lower-degree polynomial simply cannot survive $n$ derivatives, so it is automatically orthogonal to $H_n$ .

Step 4. For $m=n$ , $\partial_x^n H_n=n!\times(\text{leading coeff})=2^n n!$ , a constant, so the integral is $2^n n!\int e^{-x^2}\,\mathrm{d}x=2^n n!\sqrt\pi$ , using $\int_{-\infty}^{\infty}e^{-x^2}\,\mathrm{d}x=\Gamma(\tfrac12)=\sqrt\pi$ . Why: the factorial comes from stripping a degree- $n$ polynomial to its top term, and $\sqrt\pi$ is the Gamma function at one-half — the Gaussian integral that anchors the whole family. *As a direct check for $n=2$ : $\int(4x^2-2)^2e^{-x^2}\mathrm{d}x=8\sqrt\pi=2^2\cdot2!\cdot\sqrt\pi$ , confirming the general norm formula $2^n n!\sqrt\pi$ .*

Go deeper · Derivation C · 1 minRecurrence, derivative relation, and the ladder operators

Derivation CRecurrence, derivative relation, and the ladder operators

This derivation shows that the three-term recurrence (4) and the derivative relation (5) are not independent inputs but two differentiations of the one generating function, and then reinterprets both as the oscillator's raising and lowering operators — so the "recurrence" a student memorizes and the "ladder algebra" a physicist wields are the same equations in two languages. The plan: differentiate $G(x,t)=e^{2xt-t^2}$ once in $t$ and once in $x$ , match powers of $t$ , then translate the index shifts into operators acting on $\psi_n$ .

Step 1. Differentiate $G$ in $t$ : $\partial_t G=(2x-2t)G$ . Writing both sides as power series and using $\partial_t G=\sum H_n t^{n-1}/(n-1)!$ gives, on matching $t^n$ , the recurrence $H_{n+1}=2xH_n-2nH_{n-1}$ . Why: the factor $(2x-2t)$ mixes exactly three neighbouring coefficients, which is precisely the three-term structure guaranteed for any orthogonal family by the fact that $xH_n$ is degree $n+1$ .

Step 2. Differentiate $G$ in $x$ : $\partial_x G=2t\,G$ . Matching $t^n$ on $\sum H_n' t^n/n!=\sum 2H_n t^{n+1}/n!$ gives the derivative relation $H_n'=2nH_{n-1}$ . Why: differentiating in $x$ shifts the power of $t$ by one, which is what lowers the degree by one — the analytic seed of the lowering operator.

Step 3. Pass to the normalized functions $\psi_n=H_n e^{-x^2/2}/\sqrt{2^n n!\sqrt\pi}$ and define $a=\tfrac{1}{\sqrt2}(x+\partial_x)$ , $a^\dagger=\tfrac{1}{\sqrt2}(x-\partial_x)$ . Using (4) and (5) one finds $a\,\psi_n=\sqrt{n}\,\psi_{n-1}$ and $a^\dagger\psi_n=\sqrt{n+1}\,\psi_{n+1}$ . Why: this is the recurrence in disguise — the polynomial identities become the statement that $a^\dagger$ climbs the ladder and $a$ descends it, with the $\sqrt{n}$ factors carrying the factorial norms.

Step 4. Check: $a\,\psi_0=\tfrac{1}{\sqrt2}(x+\partial_x)\,\pi^{-1/4}e^{-x^2/2}=0$ and $[a,a^\dagger]=1$ , so the number operator gives $a^\dagger a\,\psi_n=n\,\psi_n$ and $E_n=\hbar\omega(n+\tfrac12)$ . Why: $a$ annihilating the ground state is why the ladder has a bottom rung, and $a^\dagger a$ recovers the same quantized energies derived analytically above — two roads to one spectrum. *Matching $t^n$ coefficients in the generating-function differentiations reproduces (4) and (5); the resulting ladder relations $a\psi_n=\sqrt n\,\psi_{n-1}$ , $a^\dagger\psi_n=\sqrt{n+1}\,\psi_{n+1}$ , and the commutator $[a,a^\dagger]=1$ constitute the standard oscillator algebra.*

Limiting cases

Case	Result	Meaning
Low orders	$H_0=1,\;H_1=2x,\;H_2=4x^2-2,\;H_3=8x^3-12x,\;H_4=16x^4-48x^2+12$	Leading coefficient $2^n$ , parity $(-1)^n$ ; these seed the recurrence (4).
Value at the origin	$H_{2m}(0)=(-1)^m\dfrac{(2m)!}{m!},\quad H_{2m+1}(0)=0$	Read from $e^{-t^2}=\sum H_n(0)t^n/n!$ ; odd polynomials vanish at $0$ by parity.
Probabilists' link	$H_n(x)=2^{n/2}\,He_n(\sqrt2\,x)$	Converts physicists' $H_n$ (weight $e^{-x^2}$ ) to the monic $He_n$ (weight $e^{-x^2/2}$ ).
Large $n$ , bulk (Plancherel–Rotach)	$\psi_n(x)\sim\left(\dfrac{2}{\pi}\right)^{1/4}\dfrac{1}{(2n)^{1/4}}\cos\!\left(\sqrt{2n+1}\,x-\dfrac{n\pi}{2}\right)$	Oscillatory for $\|x\|\ll\sqrt{2n+1}$ ; the classical turning point sits at $x\approx\sqrt{2n+1}$ , and $\|\psi_n\|^2$ tends to the classical $1/v(x)$ density (Bohr correspondence).

Common trapThe two conventions are not interchangeable

The most common and most costly Hermite error is quoting a formula in one normalization while using a polynomial from the other. The physicists' $H_n$ carries weight $e^{-x^2}$ , has leading coefficient $2^n$ , obeys $H_{n+1}=2xH_n-2nH_{n-1}$ , and $H_2=4x^2-2$ with norm $^2=2^n n!\sqrt\pi$ . The probabilists' $He_n$ carries weight $e^{-x^2/2}$ , is monic, obeys $He_{n+1}=xHe_n-nHe_{n-1}$ , and $He_2=x^2-1$ with norm $^2=n!\sqrt{2\pi}$ . They are related by $H_n(x)=2^{n/2}He_n(\sqrt2\,x)$ . Mathematica's `HermiteH` and most physics texts (Arfken, Stone & Goldbart) mean the physicists' $H_n$ ; most statistics texts and the standard-normal density mean $He_n$ . Before you trust any norm, generating function, or recurrence, first pin down which $H$ the source intends — a silent factor of $\sqrt2$ per degree is the classic way to get a "correct" derivation with wrong numbers.

Worked examples

1 · The norm of H₂ by direct integration (confidence builder)

Verify (1) for $n=2$ by evaluating $\displaystyle\int_{-\infty}^{\infty}H_2(x)^2\,e^{-x^2}\,\mathrm{d}x$ directly from the Gaussian moments, and confirm it equals $2^2\,2!\,\sqrt\pi$ .

Step 1. Square the polynomial: $H_2=4x^2-2$ , so $H_2^2=16x^4-16x^2+4$ . The integral splits into three Gaussian moments against $e^{-x^2}$ .

Step 2. Recall the moments $\int_{-\infty}^{\infty}e^{-x^2}\,\mathrm{d}x=\sqrt\pi$ , $\int x^2 e^{-x^2}\,\mathrm{d}x=\tfrac12\sqrt\pi$ , and $\int x^4 e^{-x^2}\,\mathrm{d}x=\tfrac34\sqrt\pi$ (each is $\Gamma$ of a half-integer). Then

Step 3. Assemble: $16\cdot\tfrac34\sqrt\pi-16\cdot\tfrac12\sqrt\pi+4\cdot\sqrt\pi=12\sqrt\pi-8\sqrt\pi+4\sqrt\pi=\mathbf{8\sqrt\pi}$ .

Step 4. Check against (1): $2^n n!\sqrt\pi=2^2\cdot 2!\cdot\sqrt\pi=\mathbf{8\sqrt\pi}$ . The direct integral and the master formula agree exactly, $\|H_2\|^2=8\sqrt\pi$ , and the factorial growth is visible across $\|H_n\|^2=\sqrt\pi,2\sqrt\pi,8\sqrt\pi,48\sqrt\pi$ for $n=0,1,2,3$ .

2 · Zero-point energy from the ground-state Hermite function

The oscillator ground state is $\psi_0(x)=\pi^{-1/4}e^{-x^2/2}$ (in units $m=\hbar=\omega=1$ ). Compute $\langle x^2\rangle$ and $\langle p^2\rangle$ , hence the ground-state energy, and show it saturates the uncertainty bound.

Step 1. Normalization: $\int\psi_0^2\,\mathrm{d}x=\pi^{-1/2}\int e^{-x^2}\,\mathrm{d}x=\pi^{-1/2}\sqrt\pi=1$ , consistent with (7) at $n=0$ .

Step 2. Position spread: $\langle x^2\rangle=\pi^{-1/2}\int x^2 e^{-x^2}\,\mathrm{d}x=\pi^{-1/2}\cdot\tfrac12\sqrt\pi=\mathbf{\tfrac12}$ .

Step 3. Momentum spread: with $p=-i\,\partial_x$ and $\psi_0'=-x\,\psi_0$ , $\langle p^2\rangle=\int|\psi_0'|^2\,\mathrm{d}x=\int x^2\psi_0^2\,\mathrm{d}x=\langle x^2\rangle=\mathbf{\tfrac12}$ — the Gaussian shares its spread equally between $x$ and $p$ .

Step 4. Energy: $E_0=\tfrac12\langle p^2\rangle+\tfrac12\langle x^2\rangle=\tfrac12\cdot\tfrac12+\tfrac12\cdot\tfrac12=\mathbf{\tfrac12}$ , i.e. $E_0=\tfrac12\hbar\omega$ — the zero-point energy, matching (6)–(7) at $n=0$ . The uncertainty product is $\Delta x\,\Delta p=\sqrt{\langle x^2\rangle\langle p^2\rangle}=\sqrt{\tfrac14}=\mathbf{\tfrac12}$ , saturating the Heisenberg bound $\Delta x\,\Delta p\ge\tfrac12$ . The Gaussian $\psi_0$ is the unique minimum-uncertainty state, and its irreducible spread is exactly the oscillator's zero-point energy.

Physics of the portrait

The functions carry the name of Charles Hermite, though — in a textbook case of Stigler's law — Laplace and Chebyshev had written them down first. What Hermite gave them was a home inside analysis, from which physics later claimed them wholesale.

Origins · 1 minCharles Hermite and how this was discovered

Charles Hermite

1822–1901 · French analyst who gave orthogonal polynomials, Hermitian forms, and the harmonic-oscillator eigenfunctions their modern form

Charles Hermite was one of the towering French analysts of the nineteenth century, and a rare mathematician whose name attaches to objects at the very centre of physics without his having done any physics at all. Born in 1822, a poor examination student who nevertheless corresponded with Europe's leading mathematicians as a teenager, he reached his deepest result in 1873 by proving that $e$ is transcendental — the first proof that a naturally occurring constant lies beyond all algebraic equations, and the template Lindemann adapted to settle $\pi$ and the squaring of the circle. He also gave his name to Hermitian matrices and forms, whose real eigenvalues are the bedrock of quantum observables. The polynomials arose in his 1864 study of a new class of series expansions; characteristically he saw them less as special functions than as an instance of a general orthogonality principle, and at once generalized them to several variables. He never lived to see their destiny: only with Schrödinger's wave mechanics in 1926 did $H_n$ become the eigenfunctions of the quantum harmonic oscillator, and only later did Wiener make them the natural eigenbasis of the Fourier transform. The same abstract structures Hermite prized — orthogonality and Hermitian symmetry — turned out to govern the physical world.

History

1810

— Laplace writes the polynomials down implicitly in his probability theory

1859

— Chebyshev studies the orthogonal family and its weight in detail

1864

— Hermite publishes the polynomials and their multivariate generalization

1926

— Schrödinger's wave mechanics makes Hₙ the harmonic-oscillator eigenfunctions

1933

— Wiener uses the Hermite functions as the eigenbasis of the Fourier transform

Going further

Texts. Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., ch. 18 (Hermite functions) · Stone & Goldbart, Mathematics for Physics, ch. on Sturm–Liouville theory and special functions · Hassani, Mathematical Physics, ch. on classical orthogonal polynomials · Byron & Fuller, Mathematics of Classical and Quantum Physics, §5–6 · Nakahara, Geometry, Topology and Physics (for the oscillator algebra in context) · Abramowitz & Stegun, Handbook of Mathematical Functions, ch. 22 · NIST DLMF, ch. 18. Key anchor results: $H_0$ through $H_4$ from recurrence (4); the norm of $H_2$ by Gaussian moments ( $=8\sqrt\pi$ , Worked Example 1); the special values $H_{2m}(0)=(-1)^m(2m)!/m!$ ; the oscillator reduction $\psi=He^{-x^2/2}\Rightarrow H''-2xH'+(\varepsilon-1)H=0$ with $\varepsilon-1=2n$ ; the ground-state result $\langle x^2\rangle=\langle p^2\rangle=\tfrac12$ , $E_0=\tfrac12$ (Worked Example 2); and the convention map $H_n(x)=2^{n/2}He_n(\sqrt2\,x)$ .

Prerequisites. Orthogonal Polynomials: General Theory · Generating Functions for Special Functions · Recurrence Relations for Special Functions · The Gamma Function. Leads to. Laguerre Polynomials · The Quantum Harmonic Oscillator · The Fourier Transform.

Ask the tutor: Ask me to build any $H_n$ from the recurrence, run a Gauss–Hermite quadrature, or walk through how peeling off the Gaussian turns the oscillator into Hermite's equation and quantizes the energy.

Why does the Gaussian weight e^(−x²) show up, and not some other decay?

Because it is the unique weight for which the family stays closed under differentiation on the whole real line. Hermite polynomials live on all of $\mathbb{R}$ , so their weight must decay faster than any polynomial grows — ruling out algebraic weights. Among rapidly decaying weights, $e^{-x^2}$ is singled out by Pearson's classification: it is the one whose logarithmic derivative $w'/w=-2x$ is linear, exactly the condition that makes Rodrigues produce polynomials and $H_n'$ proportional to $H_{n-1}$ . Physically the same $e^{-x^2}$ is the oscillator ground-state Gaussian, so the weight and the lowest wavefunction are one object.

What is the difference between a Hermite polynomial and a Hermite function?

The polynomial $H_n(x)$ grows without bound as $|x|\to\infty$ and is not square-integrable on its own. The Hermite function $\psi_n(x)=H_n(x)e^{-x^2/2}/\sqrt{2^n n!\sqrt\pi}$ multiplies it by the Gaussian envelope, forcing decay and making $\psi_n$ a genuine element of $L^2(\mathbb{R})$ . The polynomials are orthogonal only with the weight $e^{-x^2}$ in the integral; the functions are orthonormal with no weight, because that weight has been split as $e^{-x^2}=e^{-x^2/2}\cdot e^{-x^2/2}$ , one factor to each function. Physics uses $\psi_n$ ; the recurrence algebra is cleaner on $H_n$ .

How is the three-term recurrence related to the ladder operators?

They are the same equations in different notation. Differentiating the generating function gives $H_{n+1}=2xH_n-2nH_{n-1}$ and $H_n'=2nH_{n-1}$ ; rewritten for the normalized $\psi_n$ with $a=\tfrac{1}{\sqrt2}(x+\partial_x)$ and $a^\dagger=\tfrac{1}{\sqrt2}(x-\partial_x)$ , they become $a\psi_n=\sqrt n\,\psi_{n-1}$ and $a^\dagger\psi_n=\sqrt{n+1}\,\psi_{n+1}$ . So "multiply by $2x$ and subtract the neighbour" is the physicist's "apply the raising operator." Tabulating polynomials and building oscillator states are one algebra — which is why the oscillator is exactly solvable.

Why are Hermite functions the eigenfunctions of the Fourier transform?

Because the Fourier transform commutes with the operator $-\partial_x^2+x^2$ , whose eigenfunctions are exactly the $\psi_n$ . The transform turns $x$ into $i\partial_k$ and $\partial_x$ into $ik$ , leaving $-\partial^2+x^2$ invariant; any operator commuting with $\mathcal{F}$ shares its eigenbasis, so the oscillator Hamiltonian's eigenstates are automatically Fourier eigenstates. The eigenvalue is $(-i)^n$ , periodic mod 4, so $\mathcal{F}^4=1$ . The $n=0$ case is the statement that the Gaussian is its own Fourier transform; the higher $\psi_n$ continue that self-similarity up the ladder, and they underlie the fractional Fourier transform used in optics and signal processing.

What do the zeros of Hₙ have to do with numerical integration?

Everything, through Gauss–Hermite quadrature. The $n$ real simple zeros $x_1,\dots,x_n$ of $H_n$ are the optimal nodes for $\int_{-\infty}^{\infty}f(x)e^{-x^2}\,\mathrm{d}x\approx\sum_i w_i f(x_i)$ : with weights $w_i=\dfrac{2^{n-1}n!\sqrt\pi}{n^2\,[H_{n-1}(x_i)]^2}$ , the rule is exact for every polynomial of degree up to $2n-1$ — twice the accuracy of a naive $n$ -point scheme. That is why placing nodes at orthogonal-polynomial zeros is the standard route to high-accuracy integration against a Gaussian, ubiquitous in quantum expectation values and partition sums.

In what sense is "H is a polynomial" the same as "energy is quantized"?

After peeling off $e^{-x^2/2}$ , the oscillator equation becomes $H''-2xH'+(\varepsilon-1)H=0$ with $\varepsilon=2E/\hbar\omega$ . A power-series solution gives a two-term recursion whose coefficients, if the series never stops, grow like those of $e^{x^2}$ — so $\psi=He^{-x^2/2}$ would diverge and cannot be physical. The only way to keep $\psi$ normalizable is for the series to terminate into a polynomial, which happens for exactly one discrete set of eigenvalues, $\varepsilon-1=2n$ , i.e. $E_n=\hbar\omega(n+\tfrac12)$ . There is no separate "quantization postulate": demanding a well-behaved wavefunction is demanding a polynomial, and the allowed energies are the degrees at which that polynomial exists.