Introduction to Hermite Polynomials

Hermite polynomials form one of the classical families of orthogonal polynomials. They arise naturally in probability theory, numerical analysis, and quantum mechanics. Two slightly different conventions exist: the physicists' Hermite polynomials $H_n$ and the probabilists' Hermite polynomials ${\mathrm{He}}_n$. The physicists' version is commonly used in solving the quantum harmonic oscillator, while the probabilists' version appears in expansions of Gaussian random variables. Despite the difference in normalization, both satisfy recurrence relations and differential equations that reveal their deep structure.

Recurrence Relations

For the physicists' polynomials, the recurrence is $H_{n+1}=2x{H}_n-2n{H}_{n-1}$. The starting values are $H_0=1$ and $H_1=2x$. Probabilists use ${\mathrm{He}}_{n+1}=x{\mathrm{He}}_n-n{\mathrm{He}}_{n-1}$. Both satisfy a second-order differential equation, but with slightly different coefficients.

Differential Equation and Orthogonality

Physicists' Hermite polynomials solve $y^{\text{'}\text{'}}-2x{y}^{\text{'}}+2ny=0$. Probabilists' polynomials satisfy $y^{\text{'}\text{'}}-x{y}^{\text{'}}+ny=0$. Both families are orthogonal with respect to a Gaussian weight. For example, the physicists' form obeys

$\frac{1}{\sqrt{\mathrm{\pi }}}\underset{-\infty }{\overset{\infty }{\int }}=2nn!$

which means they provide an orthogonal basis for functions with Gaussian weighting. This property is invaluable when solving the quantum harmonic oscillator, where wave functions factor into a Gaussian times a Hermite polynomial.

Generating Function

A compact way to describe all Hermite polynomials is via the generating function. For physicists' polynomials, the generating function is

$e\left(2xt-t^2\right)=\sum _n^0{H}_n\left(x\right)\frac{t^n}{n}$

From this expression, you can derive recurrence relations, identities, and approximate asymptotic behavior. The generating function demonstrates the deep connection between Hermite polynomials and the Gaussian exponential.

Applications in Physics and Probability

In quantum mechanics, wave functions of the harmonic oscillator are products of a Gaussian factor and a Hermite polynomial. The energy levels correspond to the polynomial degree. In probability theory, Hermite polynomials appear in the Edgeworth expansion, providing corrections to the central limit theorem. They also underlie techniques in numerical integration called Gaussian quadrature. Because of these connections, mastering Hermite polynomials opens the door to a wide range of analytical tools.

Using the Calculator

Choose whether you want the physicists' or probabilists' version, then enter an integer degree $n$ and a value $x$. The script evaluates the polynomial via the appropriate recurrence relation. For small degrees you can verify the results manually: $H_2\left(x\right)=4x^2-2$ and ${\mathrm{He}}_2\left(x\right)=x^2-1$. Higher degrees quickly produce large integer coefficients, showcasing the complexity hidden inside this elegant family.

Why Two Conventions?

The difference between physicists' and probabilists' Hermite polynomials boils down to scaling. The probabilists' version is rescaled so that ${\mathrm{He}}_{\mathrm{n}}\left(x\right)=\frac{H_{\mathrm{n}}}{\left(\frac{x}{\sqrt{2}}\right)}{2}^{\mathrm{n}}$. This scaling keeps the variance of ${\mathrm{He}}_n$ under a standard normal distribution equal to $n!$. Both conventions satisfy orthogonality with respect to a Gaussian, but the normalization constant differs.

Historical Notes

Charles Hermite first studied these polynomials in the nineteenth century while investigating solutions to the heat equation. Later, they gained prominence through physicists analyzing oscillators and mathematicians developing probability theory. The interplay between these fields illustrates how special functions often bridge disparate areas of research. Today Hermite polynomials remain a staple in textbooks, providing accessible examples of orthogonal polynomials and their many uses.

Exploration Ideas

Try computing Hermite polynomials for various degrees and plotting them to see how their oscillations grow with $n$. Because the zeros interlace and move outward, they reveal subtle details about approximation theory. In numerical methods, these zeros serve as quadrature nodes for integrating functions with Gaussian weight. The more you experiment, the more you will appreciate how these polynomials shape countless analytical techniques.

Numerical Stability

When evaluating Hermite polynomials of high degree, the coefficients grow rapidly, leading to large intermediate values. Direct computation may suffer from round-off errors. Stable algorithms often rescale the variables or use extended precision arithmetic. Our calculator demonstrates the basic recurrence without such refinements, so extreme inputs can lead to inaccurate results. Understanding these numerical issues is crucial in applications like quantum chemistry where very high-degree polynomials arise.

Connections to Other Polynomials

Hermite polynomials belong to the wider family of hypergeometric functions. They can be expressed in terms of the confluent hypergeometric function $U$ or as limits of Jacobi polynomials. These relationships illuminate how seemingly different special functions share underlying structures. Exploring such links can help you transition from Hermite polynomials to Laguerre, Chebyshev, or Legendre families, each with its own domain of applications.