Integral of relative distance–dependent potential

[Isotropicaf]

Homework Statement

Hello,
Imagine the Hamiltonian of a two-atom molecule, you have the kinetic energy of one+the other and then you have a relative distance potential ( V(|r2-r1|^2), ri is 3 dimensional).
How to change the variable to solve the following integral with infinite limits?

Relevant Equations

Intg( exp( |r2-r1|^2) dr1dr2)

I think its going to be intg(dr2)intg(exp(r^2) dr) or something like that.

 

[Abhishek11235]

Try putting one variable on the axis

 

[Isotropicaf]

Try putting one variable on the axis

Im sorry, i don't know what you mean by that, you mean i should assume r1 as a constant and analyse how |r2-r1|^2 behaves in this condition ?

 

[Abhishek11235]

you mean i should assume r1 as a constant and analyse how |r2-r1|^2 behaves in this condition ?

No!

$|r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta}$

Now the integrations are easy!

 

[Isotropicaf]

No!

$|r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta}$

Now the integrations are easy!

Oh thanks that totally solved my problem, seems obvious now ahah just to check, the integral of the sum is the sum of the integrals?