Volume integral over a gradient (quantum mechanics)

[astrocytosis]

Homework Statement

1) Calculate the density of states for a free particle in a three dimensional box of linear size L.

2) Show that $\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x$ provided that $lim_{r \rightarrow \inf} [f(x)g(x)]=0$

3) Calculate the integral $\int xe^{-i\mathbf{q \cdot x}} e^{-Zr/a_0} \, d^3 x$

Homework Equations

Not sure

The Attempt at a Solution

[/B]
(1) seems straightforward; I followed the method in Sakurai and got $$\frac{m^{3/2}L^3 \sqrt{E}}{\pi^2 \hbar^2 \sqrt{2}}. $$ (2) and (3) though I'm confused about the context. am I supposed to be thinking of some specific functions when it comes to f and g? the limit is in r but the functions are of x? am I missing some identities that would help me solve it?

(3) doesn't look difficult but the professor said the calculation would be quite involved, so I'm clearly not thinking about it correctly, and I'm not seeing how the result (2) would help...

 

[astrocytosis]

OK, I think all I need for (2) is the product rule since f*g is going to zero, but I'm still not sure how this relates to (3)

 

[Markus Kahn]

Hi astrocytosis,

I didn't check (1) so I can't really comment on that.

(2) To see this consider the follwoing. Let $\bf v$ be a vector field and $u$ a scalar field. Now one can prove that the following identity holds
$$
\mathop{\rm div}({\bf v}u)=
(\mathop{\rm div}{\bf v})\,u+
{\bf v}\cdot\mathop{\rm grad}u.
$$
Now let $G$ be a finite domain. We than have
$$
\int_{\partial G}{\bf v}\,u\;{\rm d}\boldsymbol\sigma
=\int_G\mathop{\rm div}({\bf v}u)\;{\rm d}^3y
=\int_G(\mathop{\rm div}{\bf v})\,u\;{\rm d}^3y+
\int_G{\bf v}\cdot\mathop{\rm grad}u\;{\rm d}^3y
$$
Now if we let $G\longrightarrow \mathbb{R}^3$ and the limit you were given holds the integral over $\partial G$ vanishes and you get the desired identity.