# Integral of relative distance–dependent potential

• Isotropicaf
In summary, the conversation suggests using the equation |r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta} for simplifying the integration process. The conversation also confirms that the integral of the sum is equal to the sum of the integrals.
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.

Try putting one variable on the axis

Abhishek11235 said:
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 ?

Isotropicaf said:
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!

Abhishek11235 said:
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?

## 1. What is the concept of "Integral of relative distance-dependent potential"?

The integral of relative distance-dependent potential is a mathematical concept used in physics and chemistry to calculate the energy associated with the interaction between two particles at a given distance. It takes into account the distance between the particles and the strength of their interaction.

## 2. How is the integral of relative distance-dependent potential calculated?

The integral is calculated by integrating the relative distance-dependent potential function over the entire distance range between the two particles. This involves finding the anti-derivative of the potential function and evaluating it at the given distance limits.

## 3. What is the significance of the integral of relative distance-dependent potential?

The integral provides a measure of the total energy associated with the interaction between two particles. It can be used to understand the stability of a system and predict the behavior of particles at different distances.

## 4. How does the integral of relative distance-dependent potential relate to other mathematical concepts?

The integral of relative distance-dependent potential is closely related to the concept of potential energy, which is the energy associated with the position of particles in a system. It is also related to the concept of force, as the derivative of the potential function gives the force between the particles.

## 5. Can the integral of relative distance-dependent potential be applied to all types of interactions?

Yes, the concept can be applied to any type of interaction between two particles, such as gravitational, electrostatic, or van der Waals interactions. However, the specific form of the potential function may vary depending on the type of interaction being studied.

Replies
8
Views
892
Replies
0
Views
213
Replies
19
Views
750
Replies
1
Views
2K
Replies
5
Views
1K
Replies
1
Views
660
Replies
2
Views
2K
Replies
2
Views
830