Hermite Polynomials

Hermite polynomials are the polynomial factors in the one-dimensional quantum harmonic oscillator eigenfunctions. They also form, after multiplication by a Gaussian, a complete orthonormal basis of $L^2(\mathbb R)$.

This page uses the physicists’ convention,

$$H_0(x)=1, \qquad H_1(x)=2x,$$

which is the convention used in standard harmonic-oscillator wavefunctions. Some probability and statistics books use a different normalization, usually written $\operatorname{He}_n(x)$. Mixing the two conventions is the most common source of factor-of-two errors.

Rodrigues Formula

The physicists’ Hermite polynomial of degree $n$ is

$$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}, \qquad n=0,1,2,\ldots.$$

The first few are

$$\begin{aligned} H_0(x)&=1,\\ H_1(x)&=2x,\\ H_2(x)&=4x^2-2,\\ H_3(x)&=8x^3-12x. \end{aligned}$$

The leading term is

$$H_n(x) = 2^n x^n+\text{lower powers}.$$

Generating Function

An equivalent definition is the generating function

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

This formula is often the quickest way to derive identities. For example, differentiating with respect to $x$ gives

$$\frac{dH_n}{dx} = 2nH_{n-1}, \qquad n\ge1.$$

Differentiating with respect to $t$ gives the three-term recurrence

$$H_{n+1}(x) = 2xH_n(x)-2nH_{n-1}(x), \qquad n\ge1.$$

These relations are useful when evaluating matrix elements in the oscillator basis, deriving ladder identities, and checking numerical wavefunctions.

Differential Equation

Hermite polynomials satisfy Hermite’s differential equation

$$\frac{d^2H_n}{dx^2} -2x\frac{dH_n}{dx} +2nH_n =0.$$

In Sturm–Liouville form this can be written as

$$- \frac{d}{dx} \left( e^{-x^2} \frac{dH_n}{dx} \right) = 2n e^{-x^2}H_n(x).$$

The weight $e^{-x^2}$ is therefore not optional. It is part of the orthogonality structure.

Orthogonality

The Hermite polynomials are orthogonal on the real line with Gaussian weight:

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

This is polynomial orthogonality, not ordinary unweighted orthogonality. The factor on the right fixes the standard normalization.

The underlying Gaussian integral and half-integer normalization constants are summarized in Gamma and Beta Functions.

For $m\ne n$, the integral vanishes. For $m=n$, the norm is

$$\int_{-\infty}^{\infty} e^{-x^2} H_n(x)^2\,dx = \sqrt\pi\,2^n n!.$$

Parity and Zeros

Hermite polynomials have definite parity:

$$H_n(-x)=(-1)^nH_n(x).$$

Thus $H_n$ is even for even $n$ and odd for odd $n$. The polynomial $H_n$ has $n$ simple real zeros. In the oscillator, those zeros become the nodes of the $n$th energy eigenfunction.

Hermite Functions

The normalized Hermite functions are

$$\varphi_n(x) = \frac{1}{ \pi^{1/4}\sqrt{2^n n!} } H_n(x)e^{-x^2/2}.$$

They obey

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

They form a complete orthonormal basis of $L^2(\mathbb R)$. Thus suitable square-integrable functions can be expanded as

$$f = \sum_{n=0}^{\infty} c_n\varphi_n, \qquad c_n = \int_{-\infty}^{\infty} \varphi_n(x)^*f(x)\,dx,$$

with convergence understood in the Hilbert-space norm when $f\in L^2(\mathbb R)$.

The distinction between $H_n$ and $\varphi_n$ matters:

• $H_n$ is a polynomial and is not square-integrable on its own.
• $\varphi_n$ includes the Gaussian factor and is square-integrable.
• The oscillator wavefunction is a scaled Hermite function.

Harmonic-Oscillator Connection

For the one-dimensional harmonic oscillator, define the oscillator length

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

and the dimensionless coordinate

$$\xi=\frac{x}{\ell}.$$

The normalized energy eigenfunctions are

$$\psi_n(x) = \frac{1}{\sqrt{\ell}}\, \varphi_n\left(\frac{x}{\ell}\right).$$

Equivalently,

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

The factor $\ell^{-1/2}$ is required because

$$dx=\ell\,d\xi.$$

The oscillator differential-equation derivation explains why the acceptable power series terminates and why the energy becomes

$$E_n=\hbar\omega\left(n+\frac12\right).$$

That derivation belongs in Differential-Equation Solution. This page owns the Hermite-polynomial identities used by that solution.

Ladder Identities for Hermite Functions

Define the dimensionless lowering and raising operators

$$a = \frac{1}{\sqrt2} \left( x+\frac{d}{dx} \right), \qquad a^\dagger = \frac{1}{\sqrt2} \left( x-\frac{d}{dx} \right).$$

On Hermite functions,

$$a\varphi_n = \sqrt n\,\varphi_{n-1}, \qquad a^\dagger\varphi_n = \sqrt{n+1}\,\varphi_{n+1}.$$

Equivalently,

$$\left( x+\frac{d}{dx} \right)\varphi_n = \sqrt{2n}\,\varphi_{n-1},$$

and

$$\left( x-\frac{d}{dx} \right)\varphi_n = \sqrt{2(n+1)}\,\varphi_{n+1}.$$

These formulas are the function-space version of the harmonic-oscillator ladder algebra.

Convention Warning

The probabilists’ Hermite polynomials are commonly defined by

$$\operatorname{He}_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}.$$

They are related to the physicists’ convention by

$$\operatorname{He}_n(x) = 2^{-n/2} H_n\left(\frac{x}{\sqrt2}\right),$$

or equivalently

$$H_n(x) = 2^{n/2} \operatorname{He}_n(\sqrt2\,x).$$

The harmonic oscillator pages use $H_n$, not $\operatorname{He}_n$.

Where Hermite Functions Reappear

Hermite functions are useful beyond the elementary oscillator:

• as a basis for variational and numerical calculations on the real line;
• as benchmark eigenfunctions for spectral and finite-difference methods;
• in Gaussian wave-packet and coherent-state calculations;
• in oscillator-mode decompositions for phonons, photons, and fields;
• as eigenfunctions of the unitary Fourier transform, up to convention-dependent phases.

With the convention

$$(\mathcal F f)(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}f(x)\,dx,$$

one has

$$\mathcal F\varphi_n = (-i)^n\varphi_n.$$

If a different Fourier convention is used, the statement must be translated accordingly; see Fourier Transform Conventions.

Common Mistakes

• Confusing physicists’ $H_n$ with probabilists’ $\operatorname{He}_n$.
• Forgetting the Gaussian weight $e^{-x^2}$ in polynomial orthogonality.
• Treating $H_n$ itself as a normalizable wavefunction.
• Dropping the scale factor $\ell^{-1/2}$ when passing from $\varphi_n(\xi)$ to $\psi_n(x)$.
• Writing the oscillator Gaussian as $e^{-x^2/\ell^2}$ instead of $e^{-x^2/(2\ell^2)}$ in the wavefunction.
• Assuming all Fourier-transform phase statements are convention-free.

Cross-Links

• Quantum Harmonic Oscillator
• Differential-Equation Solution
• Hermite Functions
• Ladder-Operator Solution: First Encounter
• Harmonic Oscillator Spectrum
• Ordinary Differential Equations
• Sturm–Liouville Theory
• Orthogonal Polynomials
• L2 Spaces
• Fourier Transform
• Gamma and Beta Functions
• Fourier Transform Conventions

References

• NIST Digital Library of Mathematical Functions, Chapter 18, Orthogonal Polynomials.
• F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, 1965.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Use Rodrigues’ formula to compute $H_0$, $H_1$, and $H_2$.

Solution

For $n=0$,

$$H_0(x)=1.$$

For $n=1$,

$$H_1(x) = -e^{x^2}\frac{d}{dx}e^{-x^2} = 2x.$$

For $n=2$,

$$\frac{d^2}{dx^2}e^{-x^2} = (4x^2-2)e^{-x^2},$$

so

$$H_2(x)=4x^2-2.$$

2. Use the recurrence relation to compute $H_3$ from $H_2$ and $H_1$.

Solution

With

$$H_{n+1}=2xH_n-2nH_{n-1},$$

set $n=2$:

$$H_3 = 2xH_2-4H_1 = 2x(4x^2-2)-4(2x) = 8x^3-12x.$$

3. Derive the parity relation $H_n(-x)=(-1)^nH_n(x)$ from the generating function.

Solution

The generating function gives

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

But

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

Matching powers of $t$ gives

$$H_n(-x)=(-1)^nH_n(x).$$

4. Show that the normalized Hermite functions are orthonormal if the Hermite-polynomial orthogonality formula is assumed.

Solution

Using

$$\varphi_n(x) = \frac{1}{ \pi^{1/4}\sqrt{2^n n!} } H_n(x)e^{-x^2/2},$$

one finds

$$\int_{-\infty}^{\infty} \varphi_m(x)^*\varphi_n(x)\,dx = \frac{ \int_{-\infty}^{\infty} e^{-x^2}H_m(x)H_n(x)\,dx }{ \pi^{1/2} \sqrt{2^m m!2^n n!} }.$$

The numerator is $\sqrt\pi\,2^n n!\,\delta_{mn}$. If $m\ne n$, the result is zero. If $m=n$, the denominator is $\sqrt\pi\,2^n n!$, so the result is one.

5. Starting from $\psi_n(x)=\ell^{-1/2}\varphi_n(x/\ell)$, derive the usual normalized oscillator wavefunction.

Solution

Substitute $\xi=x/\ell$ into

$$\varphi_n(\xi) = \frac{1}{ \pi^{1/4}\sqrt{2^n n!} } H_n(\xi)e^{-\xi^2/2}.$$

Then

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

Since

$$\frac{1}{\sqrt{\ell}\,\pi^{1/4}} = \left(\frac{1}{\pi\ell^2}\right)^{1/4},$$

this is

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

6. Rewrite Hermite’s differential equation in Sturm–Liouville form.

Solution

Start from

$$H_n''-2xH_n'+2nH_n=0.$$

Multiplying by $e^{-x^2}$ gives

$$e^{-x^2}H_n'' -2xe^{-x^2}H_n' +2ne^{-x^2}H_n =0.$$

The first two terms combine as

$$\frac{d}{dx} \left( e^{-x^2}H_n' \right) = e^{-x^2}H_n'' -2xe^{-x^2}H_n'.$$

Therefore

$$- \frac{d}{dx} \left( e^{-x^2}H_n' \right) = 2ne^{-x^2}H_n.$$

This is a singular Sturm–Liouville form on the real line with weight $e^{-x^2}$.