# Hermite polynomial and transformation

• KFC
In summary, In the chapter of quantum harmonic oscillator, we use the Hermite polynomial a lot. The Fourier transformation of Hermite polynomial in wavenumber space is given by (-i)^n \exp (-k^2/2) H_n(k). However, the speaker is looking for a similar result in terms of momentum p, which is related to wavenumber by p = \hbar k. The speaker is struggling to transform the above result into one written in terms of p. The listener suggests that for a product of two functions of x, the Fourier transformation should result in a convolution in k space, but in this special case of Hermite polynomial, the equation is a simple product in k space.
KFC
In the chapter of quantum harmonic oscillator, we use the Hermite polynomial a lot. And the Fourier transformation of Hermite polynomial (in wavenumber space) gives

$$\mathcal{F} \left\{ \exp (-x^2/2) H_n(x) \right\} = (-i)^n \exp (-k^2/2) H_n(k)$$

Now I need to find the similar result in terms of momentum p, I know the relation between wavenumber and momentum is

$$p = \hbar k$$

But I still cannot transform above result to that written in terms of p. Any clue?

If you FT a product of two functions of x, you should get a convolution in k space, not a simple product.

clem said:
If you FT a product of two functions of x, you should get a convolution in k space, not a simple product.

But I am talking about a special case: Hermite polynomial, so in this case, the above equation is correct. They are just simple product in k space

## 1. What are Hermite polynomials?

Hermite polynomials are a set of orthogonal polynomials, named after French mathematician Charles Hermite. They are used in mathematical analysis and physics to solve differential equations and represent special functions.

## 2. How are Hermite polynomials defined?

Hermite polynomials are defined as a sequence of polynomials that satisfy the recurrence relation Hn+1(x) = 2xHn(x) - 2nHn-1(x), with H0(x) = 1 and H1(x) = 2x.

## 3. What is the significance of Hermite polynomials?

Hermite polynomials have various applications in mathematics and physics. They are used to solve quantum harmonic oscillator problems, as well as in the method of moments for solving differential equations. They also have applications in probability theory, statistics, and signal processing.

## 4. What is a Hermite transformation?

A Hermite transformation is a linear transformation that maps a function into a set of coefficients corresponding to the Hermite polynomial expansion of that function. It is used to simplify the representation and computation of functions, particularly in the areas of differential equations and probability theory.

## 5. How is the Hermite transformation related to the Fourier transform?

The Hermite transformation is a generalization of the Fourier transform, which is used to decompose a function into a series of complex exponential functions. In contrast, the Hermite transformation decomposes a function into a series of orthogonal Hermite polynomials. Both transformations have applications in signal processing and solving differential equations.

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