Hermite polynomials are a set of orthogonal polynomials that play a significant role in various fields such as mathematics, physics, and engineering. These polynomials are named after the French mathematician Charles Hermite and are particularly known for their applications in probability theory, quantum mechanics, and numerical analysis.

Hermite polynomials are solutions to Hermite’s differential equation, which can be expressed in two main forms: the physicist’s Hermite polynomials and the probabilist’s Hermite polynomials. Hermite polynomials are powerful mathematical tools with a wide range of applications in both theoretical and applied sciences.

What are Hermite Polynomials?

Hermite polynomials are a sequence of orthogonal polynomials that arise in probability theory, physics, and numerical analysis. Hermite polynomials are particularly known for their role in solving differential equations and their applications in quantum mechanics and statistical mechanics.

Formal Definition of Hermite Polynomial

Hermite polynomials, denoted by H_n(x), are a set of orthogonal polynomials defined by the following recurrence relation:

[Tex]H_{n+1}(x) = 2xH_n(x) – 2nH_{n-1}(x)[/Tex]

For n≥1, with initial conditions H₀(x) = 1 and H₁(x) = 2x. These polynomials satisfy the following differential equation:

[Tex]H_n”(x) – 2xH_n'(x) + 2nH_n(x) = 0[/Tex]

Types of Hermite Polynomials

There are two main types of Hermite polynomials, distinguished by their normalization and application areas:

• Physicist’s Hermite Polynomials
• Probabilist’s Hermite Polynomials

Physicist’s Hermite Polynomials

Physicist’s Hermite polynomials, denoted by H_n(x), are commonly used in quantum mechanics. They are defined by the following explicit formula:

[Tex]H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right)[/Tex]

They can appear in the solutions of the problem concerning the quantum harmonic oscillator to describe the wave functions of the model. The orthogonality condition for physicist’s Hermite polynomials is given by:

[Tex]\int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = 2^n n! \sqrt{\pi} \delta_{mn}[/Tex]

Probabilist’s Hermite Polynomials

Probabilist’s Hermite polynomials, denoted by He_n(x), are used primarily in probability theory and statistics. They are defined by:

[Tex]He_n(x) = (-1)^n e^{\frac{x^2}{2}} \frac{d^n}{dx^n} \left( e^{-\frac{x^2}{2}} \right)[/Tex]

These polynomials are linked to normal distribution and are used in the context of Edgeworth series which is an approach to approximating function for probability distributions. The orthogonality condition for probabilist’s Hermite polynomials is:

[Tex]\int_{-\infty}^{\infty} He_m(x) He_n(x) e^{-\frac{x^2}{2}} \, dx = n! \sqrt{2\pi} \delta_{mn}[/Tex]

Properties of Hermite Polynomials

There are several properties of Hermite polynomials that are as follows:

Orthogonality

It is possible to note that the orthogonality property of Hermite polynomials is the cornerstone of their application. For physicist’s Hermite polynomials, the orthogonality condition is:

[Tex]\int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = 2^n n! \sqrt{\pi} \delta_{mn}[/Tex]

Here, δ_mn is the Kronecker delta, which is 1 if m=n and 0 otherwise. The weight function in this case is e^−x^2, and the integral ensures that the polynomials are orthogonal over the entire real line.

Recurrence Relations

Hermite polynomials also obey a relation of recurrence which comes in handy in obtaining higher order Hermite polynomials from the lower order ones. The recurrence relation for physicist’s Hermite polynomials is:

[Tex]H_{n+1}(x) = 2xH_n(x) – 2nH_{n-1}(x)[/Tex]

This relation helps in simplifying the computation of Hermite polynomials which is applied in many theoretical and practical circumstances.

Differential Equations

Hermite polynomials are solutions to the Hermite differential equation:

[Tex]H_n”(x) – 2xH_n'(x) + 2nH_n(x) = 0[/Tex]

This equation highlights the polynomials’ role in solving various physical problems, particularly in quantum mechanics. The differential equation form provides a direct method to derive the polynomials.

Derivation and Representation

Hermite polynomials can be derived and represented using several methods, including Rodrigues’ formula, generating functions, and series representation.

Rodrigues’ Formula

Rodrigues’ formula provides a direct method to generate Hermite polynomials. It is given by:

[Tex]H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right)[/Tex]

This formula is derived step-by-step as follows:

• Start with the exponential function e^−x2.
• Differentiate e^−x2 n times with respect to x.
• Multiply by (−1)ⁿe^x2.
• This process generates the n-th Hermite polynomial.

Generating Function

The generating function for Hermite polynomials is a powerful tool that encapsulates all the polynomials in a single expression:

[Tex]G(t, x) = e^{2xt – t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!}[/Tex]

To derive Hermite polynomials from the generating function, follow these steps:

• Expand the generating function in a Taylor series around t=0.
• The coefficient of tⁿ/n! in the expansion gives the n-th Hermite polynomial H_n(x).

Series Representation

Hermite polynomials can also be expressed as a series:

[Tex]H_n(x) = \sum_{k=0}^{\left\lfloor n/2 \right\rfloor} \frac{(-1)^k n!}{k!(n-2k)!} (2x)^{n-2k}[/Tex]

This series representation is useful for understanding the structure of Hermite polynomials and for practical computations.

Applications of Hermite Polynomials

The applications of the Hermite Polynomials are as follows:

• Quantum Mechanics: Employed in solving the Schrödinger equation quantum harmonic oscillator problems.
• Probability Theory: These are present in the context of the Edgeworth series and are connected with the normal distribution.
• Numerical Analysis: Employed in approximation methods and solving differential equations.
• Combinatorics: Applied in various counting problems and generating functions.
• Signal Processing: Utilized in algorithms for filtering and signal approximation.

Conclusion

Hermite polynomials are the class of orthogonal polynomials which are directly used in various branches of mathematics, physics, and engineering. For that reason, properties such as orthogonality, recurrence relation, differential equations, etc, make them useful instruments in problem-solving.

Read More,

• Polynomials
• Orthogonal Vectors
• Orthogonal Matrix
• Orthogonal and Orthonormal Vectors

FAQs on Hermite Polynomials

What is the significance of the Hermite polynomial?

The Hermite Polynomials are used in solving differential equations which feature in physics particularly quantum mechanics (for instance the harmonic oscillator). They are also involved in numerical computation and probability theory as well.

Are Hermite polynomials normalized?

No, Hermite Polynomials are not inherently normalized. However, they can be scaled to form normalized versions that are orthonormal under a specific weight function in a given interval.

What are the asymptotics of the Hermite function?

The asymptotics of Hermite functions describes their behavior for large indices or arguments, showing that they resemble Gaussian functions, which are significant in approximations and practical applications.

How do Hermite Polynomials relate to Gaussian quadrature?

Hermite Polynomials are used in Gaussian quadrature to provide accurate numerical integration, particularly for functions involving a Gaussian weight, optimizing the approximation of definite integrals.

Why are Hermite Polynomials orthogonal?

Hermite Polynomials are orthogonal with respect to the weight function e^-x2 over the entire real line, meaning their inner product is zero unless they are the same polynomial, simplifying many mathematical problems.