3D from of a potential

[Morberticus]

I have a nifty expression

$$\frac{1}{r} = \frac{\rm{erf}(r/\alpha)}{r} + \frac{\rm{erfc}(r/\alpha)}{r}$$

I can manipulate it easily enough in one dimension to get a similar expression for $$\frac{1}{|r|}$$.

Having a little trouble with the expression in three dimensions though$$\frac{1}{|\textbf{r}|}$$

Instinctively I want to write

$$\frac{1}{|\textbf{r}|} = \frac{\rm{erf}(|\textbf{r}|/\alpha)}{|\textbf{r}|} + \frac{\rm{erfc}(| \textbf{r} |/\alpha)}{|\textbf{r}|}$$

Can I do this? I think I can, as |r| is just a scalar, right?

Thanks

 

[jvicens]

for the question! Yes, you are correct in your instinct to write the expression in three dimensions as

\frac{1}{|\textbf{r}|} = \frac{\rm{erf}(|\textbf{r}|/\alpha)}{|\textbf{r}|} + \frac{\rm{erfc}(| \textbf{r} |/\alpha)}{|\textbf{r}|}

As you mentioned, |r| is just a scalar, so it can be treated the same way in both one and three dimensions. The only difference is that in three dimensions, |r| represents the magnitude of a vector, while in one dimension it represents the absolute value of a scalar. So, your expression is perfectly valid and can be used to manipulate and solve problems in three dimensions. Keep up the good work!