I'd like some help justifying the approximation that the vibrational and rotational motion of a diatomic molecule is separable.(adsbygoogle = window.adsbygoogle || []).push({});

For two atoms of masses m1 and m2 the full hamiltonian is

[itex]H=-\hbar ^2 /2m_1 \nabla _1 ^2 - \hbar^2 /2m_2 \nabla _2 ^2 + V(r-r_0)[/itex]

Where V(r-r0) is the potential energy function, r0 is the equilibrium bond length and r is the atomic separation [itex]r=|\textbf{x} _1- \textbf{x} _2|[/itex]. This is seperable into a centre of mass hamiltonian and one in terms of the reduced mass, in the molecular frame. The hamiltonian in the molecular frame is

[itex]H_\mu = -\hbar ^2 /2\mu \nabla ^2 - V(|\textbf{x}|-r_0)[/itex]

From this point, I'm not sure how to show that the hamiltonian is approximately separable. I tried writing it in terms of the equilibrium displacement [itex]x=r-r_0[/itex] and then by saying in the approximation that [itex]x>>r_0 , x/r_0 \approx 0 [/itex] so in the hamiltonian [itex] r^2=r_0^2(1+x/r_0)^2\approx r_0^2[/itex] but since [itex]r=x+r_0[/itex] is a variable and not a constant of motion, this seems like a dodgy approximation. Any help would be valued greatly!

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# Separability of vibrational and rotational motion for diatomics

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