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Expressing phase space differential in terms of COM

  • #1
59
1

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.
 

Answers and Replies

  • #2
205
84
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).
 

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