Expressing phase space differential in terms of COM

[ghostfolk]

Homework Statement

The Hamiltonian for a single diatomic molecule of identical atoms is given as $$H=\dfrac{\vec{p_1}\cdot\vec{p_1}}{2m}+\dfrac{\vec{p_2}\cdot\vec{p_2}}{2m}+\dfrac{K}{2}(\vec{r_1}-\vec{r_2})\cdot(\vec{r_1}-\vec{r_2})$$. Find the grand canonical partition function for a gas of such molecules, neglecting the interactions between molecules.

2. Homework Equations
$$Z=\dfrac{1}{N!}\int\dfrac{d^3p_id^3x_i}{h}e^{-\beta H}$$

The Attempt at a Solution

I know that if we can express the momenta in terms of the center of mass coordinates and the relative coordinates, $$R=(r_1+r_2)/2$$ and $$r=r_1-r_2$$. However I am not sure how to express the differential in terms of the center of mass and relative coordinates.

 

[Fred Wright]

You have the relative coordinate r = r₁-r₂ and the COM coordinate R = (r₁m₁ + r₂m₂)/(m₁+m₂). It's easy to show that:
∂ /∂r₁ = m₁/(m₁+m₂)∂ /∂R + ∂ /∂r and
∂ /∂r₂ = m₂/(m₁+m₂)∂ /∂R - ∂ /∂r.
The Hamiltonian becomes
H(r,R) = -ħ²/(2M)(∂² /∂R² )- ħ²/(2μ))(∂² /∂r²) + K/2(r⋅r)
where M = m₁+ m₂ and μ=(m₁m₂)/(m₁+ m₂).