At sufficiently high temperatures, a diatomic molecule may suffer vibration mode

Question

At sufficiently high temperatures, a diatomic molecule may suffer vibration modes of excitation above the normal rotational modes. Consider that the two atomic nuclei are bound through a central potential V(r) which has a minimum at the distance r = a. At low temperatures, the nuclei stay at this interparticle spacing, and the effective Hamiltonian is that of the rigid rotor = L2/2 mua2 + V(a) where mu is the reduced mass of the two nuclei. At higher temperatures, the particles vibrate about this minimum. The Hamiltonian becomes H = p2r/2mu + L2/2mu2r2 + V(r) This problem will ask you to explore the roto vibrational spectrum of this molecule. Expand the potential about the minimum value r = a : V(r) V(r)|r = 1 + V'(r)|r = a (r - a) + 1/2 V"(r) |r = a (r - a)2 Simplify this, You should define V(r)|r = a = -Vo and V"(r)|r = k at r = a, respectively. What simple 1-D problem does this resemble?

Explanation / Answer

When you expand this potential about the minimum, you can ignore the constant term at the beginning ( V(a)) , we know that you are expanding about the minimum, so V'(r=a) = 0 (the definition of a minimum). All that we are left with is the third term, (1/2)V"(a)(r-a)^2. What this looks like is (1/2)k(r)^2, when k = V"(a). This represents some sort of restoring potential. The simple 1-D problem this represents is the quantum harmonic oscillator.

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