What is Hermite polynomials: Definition and 34 Discussions

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:

signal processing as Hermitian wavelets for wavelet transform analysis
probability, such as the Edgeworth series, as well as in connection with Brownian motion;
combinatorics, as an example of an Appell sequence, obeying the umbral calculus;
numerical analysis as Gaussian quadrature;
physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term




{\displaystyle {\begin{aligned}xu_{x}\end{aligned}}}
is present);
systems theory in connection with nonlinear operations on Gaussian noise.
random matrix theory in Gaussian ensembles.Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new. They were consequently not new, although Hermite was the first to define the multidimensional polynomials in his later 1865 publications.

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  1. F

    I Adapting Hermite polynomials

    Hello everyone. I am working with generalized polynomial chaos. To represent a Normal random variable, the Hermite polynomials are used. However, as far as I understand, these represent N(0,1); if what I have read is correct, if I want to work with any other mean and variance, I shoud simply...
  2. R

    Normalizing Hermite Polynomials

    Homework Statement Evaluate the normalization integral in (22.15). Hint: Use (22.12) for one of the $H_n(x)$ factors, integrate by parts, and use (22.17a); then use your result repeatedly.Homework Equations (22.15) ##\int_{-\infty}^{\infty}e^{-x^2}H_n(x)H_m(x)dx = \sqrt{\pi}2^nn!## when ##n=m##...
  3. D

    A Integral involving Hermite polynomials

    Hello. I've an integral: \int_{-\infty}^{0}x\exp(-x^2)H_n(x-a)H_n(x+a)dx Of course for any given n it can be calculated, but I'm interested if there is some general formula for arbitrary n. Could someone with access type that into Mathematica? In case that there exists general formula, idea how...
  4. thecourtholio

    Using generating function to normalize wave function

    Homework Statement Prove that ##\psi_n## in Eq. 2.85 is properly normalized by substituting generating functions in place of the Hermite polynomials that appear in the normalization integral, then equating the resulting Taylor series that you obtain on the two sides of your equation. As a...
  5. weezy

    I Proving Hermite polynomials satisfy Hermite's equation

    My book (by Mary L Boas) introduces first the Hermite differential equation for Hermite functions: $$y_n'' - x^2y_n=-(2n+1)y_n$$ and we find solutions like $$y_n=e^{x^2/2}D^n e^{-x^2}$$ where ##D^n=\frac{d^n}{dx^n}## Now she says that multiplying ##y_n## by ##(-1)^ne^{x^2/2}## gives us what is...
  6. nmsurobert

    3rd and 4th hermite polynomials

    Homework Statement Calculate the third and fourth hermite polynomials Homework Equations (1/√n!)(√(mω/2ħ))n(x - ħ/mω d/dx)n(mω/πħ)1/4 e-mωx2/2ħ ak+2/ak = 2(k-n)/((k+2)(k+1)) The Attempt at a Solution i kind of understand how how to find the polynomials using the first equation up to n=1. I'm...
  7. G

    Deriving hermite differential equation from schrødinger harm oscillator

    Homework Statement I am trying to obtain the hermite polynomial from the schrødinger equation for a har monic oscillator. My attempt is shown below. Thank you! The derivation is based on this site: http://www.physicspages.com/2011/02/08/harmonic-oscillator-series-solution/ The Attempt at a...
  8. C

    Hermite Polynomials: What Are the Initial Values?

    I'm currently reading a text which uses Hermite polynomials defined in the recursive manner. The form of the polynomials are such that C0 C1 are the 0th and 1st terms of a taylor series that generate the remaining coefficients. The author then says the standard value of C1 and C0 are used, but...
  9. C

    Taylor series expansion of an exponential generates Hermite

    Homework Statement "Show that the Hermite polynomials generated in the Taylor series expansion e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞) are the same as generated in 7.58*." 2. Homework Equations *7.58 is an equation in the book "Introductory Quantum Mechanics" by...
  10. B

    Need help with Schrödinger and some integration

    My wave function: ##\psi_2=N_2 (4y^2-1) e^{-y^2/2}.## Definition of some parts in the wavefunction ##y=x/a##, ##a= \left( \frac{\hbar}{mk} \right)##, ##N_2 = \sqrt{\frac{1}{8a\sqrt{\pi}}}## and x has an arrange from ##\pm 20\cdot 10^{-12}##. Here is my integral: ##<x^2> =...
  11. G

    Can you help me prove the integral for Hermite polynomials?

    Hi. I'm off to solve this integral and I'm not seeing how \int dx Hm(x)Hm(x)e^{-2x^2} Where Hm(x) is the hermite polynomial of m-th order. I know the hermite polynomials are a orthogonal set under the distribution exp(-x^2) but this is not the case here. Using Hm(x)=(-1)^m...
  12. D

    Expansion of Cos(x) in Hermite polynomials

    [/itex]Homework Statement Find the first three coeficents c_n of the expansion of Cos(x) in Hermite Polynomials. The first three Hermite Polinomials are: H_0(x) = 1 H_1(x) = 2x H_0(x) = 4x^2-2The Attempt at a Solution I know how to solve a similar problem where the function is a polynomial of...
  13. S

    Write sin in terms of Hermite polynomials

    Homework Statement Write ##sin(ax)## for ##a \in \mathbb{R}##. (Use generating function for appropriate ##z##) Homework Equations ##e^{2xz-z^2}=\sum _{n=0}^{\infty }\frac{H_n(x)}{n!}z^n## The Attempt at a Solution No idea what to do. My idea was that since...
  14. A

    Calculating Integrals with Hermite Polynomials

    Hello , i need to calculate the following integral \int_{-\infty}^{\infty} x^4 H(x)^2 e^{-x^2} dx i tried using the recurrence relation, but i don't go the answer
  15. R

    What is the exact form of the zeros of Hermite polynomials?

    So I was working on eigenvalues of tridiagonal matrices, interestingly I get hermite polynomials as the solution. Is it possible to get an exact form for the zeros of hermite polynomials?
  16. T

    Integrals of products of Hermite polynomials

    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 \int{ H_n(x) H_m(\alpha x) dx} where H_n, H_m are Hermite polynomials?
  17. A

    Hermite polynomials and Schwartz space

    Homework Statement I'm supposed to show that the Hermite Polynomials are in Schwartz space h_n = \frac{1}{\sqrt{n!}}(A^{\dagger})^n h_0 where A^{\dagger} = \frac{1}{\sqrt{2}}(-\frac{d}{dx} + x) and h_0 = \pi^{-1/4}e^{-x^2/2} Homework Equations Seminorm...
  18. M

    Differential Equations - Hermite Polynomials

    Homework Statement Here is the entire problem set, but (obviously) you don't have to do it all, if you could just give me a few hints on where to even start, because I am completely lost. Recall that we found the solutions of the Schrodinger equations (x^2 - \partial_x ^2) V_n(x) =...
  19. T

    Visual basic algorithm for computing hermite polynomials

    Please I need Visual Basic algorithm for computing Hermite polynomials. Any one with useful info? Thanks.
  20. F

    Solutions to the Harmonic Oscillator Equation and Hermite Polynomials

    How are Hermite Polynomials related to the solutions to the Schrodinger equation for a harmonic oscillator? Are they the solutions themselves, or are the solutions to the equation the product of a Hermite polynomial and an exponential function? Thanks!
  21. Z

    Solving Hermite Polynomials: Find Form from Definition

    In a past exam paper at my uni I am asked to show that the hermite polynomials are solutions of the hermite diff. equation but first there is the following \Phi(x,t)=\exp (2xh-h^2)=\sum_{n=0}^{\infty} \frac{h^n}{n!}H_n(x) So I need to find the form of H_n first, and I'm stuck. I tried...
  22. C

    Integral involving Hermite polynomials

    Homework Statement The Hermite polynomials H_n(x) may be defined by the generating function e^{2hx-h^2} = \sum_{n=0}^{\infty}H_n(x)\frac{h^n}{n!} Evaluate \int^{\infty}_{-\infty} e^{-x^2/2}H_n(x) dx (this should be from -infinity to infinity, but for some reason the latex won't work!)...
  23. A

    Show that the Hermite polynomials H2(x) and H3(x).

    Hi guys. I am new, and i need help badly. I have been asked this question and I have no idea how to do it. Any help would be appreciated! Show that the Hermite polynomials H2(x) and H3(x) are orthogonal on x € [-L, L], where L > 0 is a constant, H2(x) = 4x² - 2 and H3(x) = 8x³ - 12x...
  24. P

    Indefinite integral (Hermite polynomials)

    Homework Statement I need to evaluate the following integral: \int_{-\infty}^{\infty}x^mx^ne^{-x^2}dx I need the result to construct the first 5 Hermite polynomials. Homework Equations The Attempt at a Solution First I tried arbitrary values for "m" and "n". I was not able to...
  25. J

    Quantum mechanics hermite polynomials

    Homework Statement Show that the one-dimensional Schr¨odinger equation ˆ (p^2/2m) ψ+ 1/2(mw^2)(x)ψ = En ψ can be transformed into (d^2/d ξ ^2)ψ+ (λn- ξ ^2) ψ= 0 where λn = 2n + 1. using hermite polynomials Homework Equations know that dHn(X)/dX= 2nHn(x) The Attempt at a...
  26. A

    Hermite Polynomials As Part of the Solution to the Harmonic Oscillator

    Homework Statement When trying to generate solutions to the harmonic oscillator, I'm trying to use hermite polynomials. I understand that there's a recursive relationship between the hermite polynomials but I'm confused in how each hermite polynomial is generated. Homework Equations...
  27. H

    Integral involving Hermite polynomials

    Hello! Is there any way of calculating the integral of H_n(x) * H_m(x) * exp(-c^2 x^2) with x going from -infinity to +infinity and c differs from unity. I'm aware that c=1 is trivial case of orthogonality but I'm really having a problem with the general case. (I should say that this isn't a...
  28. P

    Proof Normalization Hermite Polynomials

    Can anyone PROOVE how to find out the normalisation of hermite polynomial?
  29. D

    Hermite Polynomials: Spans All Polynomials f from R to R?

    Since the Hermite Polynomials are orthogonal, could one state that they span all polynomials f where f : R \rightarrow R? This would be EXTREMELY useful for the harmonic oscillator potential in quantum mechanics...
  30. MathematicalPhysicist

    How can I show the expansion of Hermite Polynomials using exponential functions?

    I need to show that: \sum_{n=0}^{\infty}\frac{H_n(x)}{n!}y^n=e^{-y^2+2xy} where H_n(x) is hermite polynomial. Now I tried the next expansion: e^{-y^2}e^{2xy}=\sum_{n=0}^{\infty}\frac{(-y)^{2n}}{n!}\cdot \sum_{k=0}^{\infty}\frac{(2xy)^k}{k!} after some simple algebraic rearrangemnets i...
  31. I

    Are Hermite Polynomials Always Cubic When Used for Interpolation?

    are hermite interpolationg polynomials necessarily cubic even when used to interpolate between two points? this page would have me believe so in calling it a "clamped cubic" : http://math.fullerton.edu/mathews/n2003/HermitePolyMod.html
  32. D

    Differential equation Hermite polynomials

    I got a problem in quantum physics that i have come to a differential equation but I don't see how to solve it, its on the form F''(x)+(Cx^2+D)F(x)=0. How should I solve it? Thanks
  33. U

    Hermite Polynomials Homework: Integral w/ p>r & p=r

    Homework Statement Show that \int_{-\infty}^{\infty} x^r e^{-x^2} H_n(x) H_{n+p} dx = 0 if p>r and = 2^n \sqrt{\pi} (n+r)! if p=r. with n, p, and r nonnegative integers. Hint: Use this recurrence relation, p times: H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)Homework Equations...
  34. N

    Proving Hermite Equation with Hermite Polynomials

    Im stuck on this question :( The Hermite polynomials can be defined through \displaystyle{F(x,h) = \sum^{\infty}_{n = 0} \frac{h^n}{n!}H_n(x)} Prove that the H_n satisfy the hermite equation \displaystyle{H''_n(x) - 2xH'_n(x) + 2nH_n(x) = 0} Using...