Fourier Expansion in 3D: Expansion of V(r)

[Logarythmic]

I'm used to use

$$\tilde{f} (x) = a_n|e_n>$$

where

$$|e_n> = e^{2 \pi inx / L}$$

and

$$a_n = \frac{1}{L}<e_n|f>$$

for my Fourier expansions.

How do I expand a function in 3 dimensions, for example

$$V(\vec{r}) = \frac{e^{-\lambda r}}{r}$$

?

 

[quasar987]

There is still just one variable in there, no?

 

[quasar987]

---I deleted this, it's mostly nonsense and it does't apply to the problem.---

But in my opinion, mathematically, it makes no difference if you have an r or an x in there; just do the Fourier expansion btw r_0 and r_1 as you would a fct of x.

 

[OlderDan]

I'm used to use

$$\tilde{f} (x) = a_n|e_n>$$

where

$$|e_n> = e^{2 \pi inx / L}$$

and

$$a_n = \frac{1}{L}<e_n|f>$$

for my Fourier expansions.

How do I expand a function in 3 dimensions, for example

$$V(\vec{r}) = \frac{e^{-\lambda r}}{r}$$

?

The basis functions are products whose factors are the 1-D functions in x, y, and z.

 

[Logarythmic]

So $$|e_n> = e^{2 \pi in \vec{r} /L} = e^{2 \pi in x /L} e^{2 \pi in y /L} e^{2 \pi in z /L}$$ and $$r = \sqrt{x^2 + y^2 + z^2}$$?

 

[OlderDan]

Yes for the last product on the right and the r; the Ls could be different for each dimension.

$$|e_n> = e^{2 \pi in x /L_x} e^{2 \pi in y /L_y} e^{2 \pi in z /L_z}$$

 

[Logarythmic]

So to expand $$V(\vec{r})$$ I have to rewrite it in terms of x, y ,z or does quasar987 have a point there?

 

[OlderDan]

So to expand $$V(\vec{r})$$ I have to rewrite it in terms of x, y ,z or does quasar987 have a point there?

I thought you wanted the expansion for any function in 3-D. If the function is only a function of r, then you could do a 1-D expansion in r. There are other orthogonal functions that are often used in 3-D in cylindrical or spherical coordinates.

 

[Logarythmic]

Yes, first of all I want to solve this problem but I also want to learn something from it. ;)