# $\int d^3\vec{r}_1\int d^3\vec{r}_2e^{-\frac{\beta K}{2}|\vec{r}_1-\vec{r}_2|^2}=V\int d^3\vec{r}e^{-\frac{\beta K}{2}|\vec{r}|^2}$

I was reading the solution of a problem when I saw this:

$$\int d^3\vec{r}_1\int d^3\vec{r}_2e^\left(-\frac{\beta K}{2}|\vec{r}_1-\vec{r}_2|^2\right)=V\int d^3\vec{r}e^\left(-\frac{\beta K}{2}|\vec{r}|^2\right)$$ where $$\int d^3\vec{r}_n=V\;\;\forall\;\;n$$ $$\beta, K \longrightarrow$$ constants

Just to give you a bit of the context, there is a step of the solution of a statistical physics problem on canonical ensemble where the solver assumpted this.

• Awful title choice. Please make the title more specific to the question. Apr 23 at 6:12
• @AdamRubinson so I put the whole integral in the title as I didn't know how to describe the problem properly
– sbb
Apr 23 at 6:20
• Have you tried a change of variables? Apr 23 at 6:53
• This is just a rotation by letting $r'=r_1+r_2$ and $r= r_1-r_2$ (or some scalar multiple thereof) but you may want to check that the integral isn't missing a constant somewhere. As written, the unscaled summations do not have Jacobian 1 Apr 23 at 7:01