Integrals of products of Hermite polynomials

In summary, Hermite polynomials are orthogonal polynomials named after Charles Hermite that are commonly used in mathematical analysis, particularly in the study of harmonic oscillators and quantum mechanics. Integrals of products of Hermite polynomials have various applications in physics and engineering, such as in the calculation of wave functions and in the solution of differential equations. They can be calculated using techniques such as generating functions, the Gram-Schmidt process, and recurrence relations. These integrals have special properties such as recurrence relations and orthogonality conditions, and have connections to other special functions. Real-world applications of integrals of products of Hermite polynomials include modeling quantum mechanical systems, solving differential equations, and calculating probabilities and moments of random variables in statistics
  • #1
tommyli
23
0
Hey people,

I need to calculate inner product of two Harmonic oscillator eigenstates with different mass. Does anybody know where I could find a formula for
[itex]
\int{ H_n(x) H_m(\alpha x) dx}
[/itex]

where [itex]H_n, H_m[/itex] are Hermite polynomials?
 
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  • #2
Here, for example:
http://www.wolframalpha.com/input/?i=Integrate[HermiteH[3%2C+x]*HermiteH[5%2C+x%2F3]%2C+x]

(use {x,-Infinity,+Infinity} for the last argument to get a definite integral, although this does not actually converge for this example)
 

FAQ: Integrals of products of Hermite polynomials

1. What are Hermite polynomials?

Hermite polynomials are a type of orthogonal polynomials that are named after the French mathematician Charles Hermite. They are commonly used in mathematical analysis, particularly in the study of harmonic oscillators and quantum mechanics.

2. What is the significance of integrals of products of Hermite polynomials?

Integrals of products of Hermite polynomials have various applications in physics and engineering, such as in the calculation of wave functions and in the solution of differential equations. They also have connections to probability theory and statistical mechanics.

3. How are integrals of products of Hermite polynomials calculated?

Integrals of products of Hermite polynomials can be calculated using various techniques, such as the method of generating functions, the Gram-Schmidt process, and the use of recurrence relations. The specific method used depends on the specific problem and context.

4. Are there any special properties of integrals of products of Hermite polynomials?

Yes, there are several special properties of integrals of products of Hermite polynomials. For example, they satisfy certain recurrence relations and orthogonality conditions, and they have connections to other types of special functions, such as Bessel functions and Laguerre polynomials.

5. What are some real-world applications of integrals of products of Hermite polynomials?

Integrals of products of Hermite polynomials have various real-world applications, such as in physics for modeling quantum mechanical systems, in engineering for solving differential equations, and in statistics for calculating probabilities and moments of random variables. They also have applications in the study of Brownian motion and other stochastic processes.

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