# Homework Help: 3rd and 4th hermite polynomials

1. Feb 9, 2016

### nmsurobert

1. The problem statement, all variables and given/known data
Calculate the third and fourth hermite polynomials

2. Relevant 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))
3. 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 not sure i want to attempt to find it with n=3 because that will make (x - ħ/mω d/dx)n raised to the 3rd power. and then the 4th power.
we are provided with the second equation but i don't understand how to use it. there is an example in the book (townsend) and does the first 3 polynomials but the example makes no sense to me.

can anyone provide me with some insight?

2. Feb 9, 2016

### SteamKing

Staff Emeritus
It's not clear where you obtained this generating formula for Hermite polynomials, nor what the difficulty is with the alternate representation.

https://en.wikipedia.org/wiki/Hermite_polynomials

You shouldn't have to split the atom to come up with these.

3. Feb 9, 2016

### nmsurobert

The difficulty with the alternate representation is I don't understand how to use it.
Apparently when n=1, the solution is X. Where does that come from?

4. Feb 9, 2016

### nmsurobert

Even using one of the other formulas on Wikipedia like

(X - d/dx)n

For n=1, d/dx = 0 so the solution is X. I see that.

But for n=2 the solution is x2 -1.

Why isn't it just x2 don't all the d/dx go to zero?

5. Feb 9, 2016

### nmsurobert

Even if I write it out as x^2 -d/dx(2x) + (d/dx)^2
I still don't get the correct solution for n=2.

6. Feb 14, 2016

### vela

Staff Emeritus
You shouldn't see that d/dx=0 because that's wrong. You need to apply the operator to $e^{-x^2/2}$ and then multiply the result by $e^{x^2/2}$. For $n=1$, you get
$$H_1(x) = e^{x^2/2} \left(x - \frac{d}{dx}\right)^1 e^{-x^2/2} = e^{x^2/2} (x e^{-x^2/2} + x e^{-x^2/2}) = 2x$$

7. Feb 15, 2016

### nmsurobert

ahhhh that makes sense. i spoke to the instructor and he cleared it up a bit for me too but he didn't explain it like that. thank you!