Hermite Polynomial
Introduction
Hermite polynomials are a classical family of orthogonal polynomials that arise in probability, combinatorics, and numerical analysis. They are named after the French mathematician Charles Hermite, who contributed significantly to their development. Hermite polynomials are solutions to the Hermite differential equation, a second-order linear differential equation. These polynomials are particularly useful in physics, especially in quantum mechanics, where they appear in the solutions of the quantum harmonic oscillator.
Definition and Properties
Hermite polynomials, denoted as \( H_n(x) \), can be defined in several ways, including through a generating function, a recurrence relation, and a differential equation. The most common definition is through the Rodrigues' formula:
\[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \]
This formula highlights the polynomials' orthogonality and their relationship to the Gaussian function. Hermite polynomials are orthogonal with respect to the weight function \( e^{-x^2} \) on the interval \((-\infty, \infty)\):
\[ \int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = \sqrt{\pi} 2^n n! \delta_{mn} \]
where \(\delta_{mn}\) is the Kronecker delta function.
Generating Function
The generating function for Hermite polynomials is given by:
\[ e^{2xt - t^2} = \sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n \]
This function is particularly useful for deriving various properties of Hermite polynomials, including their recurrence relations and explicit forms.
Recurrence Relations
Hermite polynomials satisfy the following recurrence relations:
\[ H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x) \]
\[ \frac{d}{dx} H_n(x) = 2n H_{n-1}(x) \]
These relations are instrumental in computational applications, allowing for the efficient calculation of Hermite polynomials.
Differential Equation
Hermite polynomials are solutions to the Hermite differential equation:
\[ y - 2xy' + 2ny = 0 \]
This equation is a special case of the Sturm-Liouville problem, which is fundamental in the theory of orthogonal polynomials.
Applications
Hermite polynomials have numerous applications across different fields of science and engineering.
Quantum Mechanics
In quantum mechanics, Hermite polynomials appear in the wave functions of the quantum harmonic oscillator. The solutions to the Schrödinger equation for a harmonic oscillator involve Hermite polynomials, which describe the oscillator's energy eigenstates. The wave functions are given by:
\[ \psi_n(x) = \left( \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} \right) e^{-x^2/2} H_n(x) \]
These functions form an orthonormal basis for the space of square-integrable functions, which is essential for quantum state analysis.
Probability and Statistics
In probability theory, Hermite polynomials are used in the context of the Edgeworth series, which provides an asymptotic expansion of probability distributions. They also play a role in the study of Gaussian processes and in the construction of Wiener chaos.
Numerical Analysis
Hermite polynomials are employed in numerical analysis, particularly in Gaussian quadrature, where they are used to approximate integrals of functions with Gaussian weight functions. This method is known as Hermite-Gauss quadrature and is crucial for efficiently evaluating integrals in computational applications.
Orthogonality and Completeness
The orthogonality of Hermite polynomials is a key property that makes them useful in various applications. The polynomials form a complete orthogonal system with respect to the weight function \( e^{-x^2} \), meaning any function satisfying certain conditions can be expressed as a series of Hermite polynomials. This property is exploited in Fourier-Hermite series expansions, which are used in solving partial differential equations and in signal processing.
Hermite Functions
Hermite functions are related to Hermite polynomials and are defined as:
\[ \phi_n(x) = \left( \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} \right) e^{-x^2/2} H_n(x) \]
These functions are eigenfunctions of the Fourier transform and are used in various applications, including quantum optics and signal processing. The Hermite functions form an orthonormal basis for the space of square-integrable functions, similar to the role of Hermite polynomials in polynomial spaces.
Generalizations and Extensions
Hermite polynomials have been generalized in various ways to extend their applicability. One such generalization is the generalized Hermite polynomials, which are used in the study of non-standard weight functions. Another extension is the q-Hermite polynomials, which arise in the context of quantum groups and q-calculus.
Historical Context
The study of Hermite polynomials dates back to the 19th century, with contributions from Charles Hermite and other mathematicians such as Joseph Fourier and Pierre-Simon Laplace. The development of these polynomials was driven by the need to solve differential equations and to understand the properties of functions in mathematical physics.