Question

5. A diatomic molecule (like H2) can be modeled as two atoms of equal mass m, connected by a rigid massless rod of length a. The system is free to rotate in 3-D. I claim the moment of inertia of this molecule around its ceater of mass is a. (Feel free to convince yourself that factor of k is coect!) Big hint if you 're having trouble getting started: this problem is directly related to McIntyre's Ch A) The energy of this system is JUST rotational kinetic energy, nothing more. Express the classical energy in terms of the angular momentum of the system (and the quantities "mand "a) Given that, write down the quantum Hamiltonian (This is pretty much just a "write it down" question what's kinetic energy for a rotating object? It was answered as a clicker question at the start of class one day last week) n2n(n+1) Use this to show that the allowed energies of this quantum system are En deduce the degeneracy of each of the energy eigenstates. (Hnt: Asking about degeneracy means tell me, for a given energy, how many distinct (linearly independent) eigenstates you could have. A degeneracy of "1" means there is I state per energy, i.e. no degeneracy at all. That will generally not be the case here.) ,n= 0 1, 2, Then na B) The two atoms in a hydrogen molecule are about .7 A apart. If you could measure the rotational spectrum energies for Hydrogen molecules, what is the lowest possible value you would expect to see, in eV? Remember that measured energies are always the difference of allowed energy eigenvalues! i) What is the wavelength of the light you would see? ii) What part of the EM spectrum is this? i) What is the next higher possible observable transition energy (in eV)? Note: It hrns out for a subtle reason involvïng the symmehy of the 2 hydrogens, which results n zero electric dipole moment, that this transition is forbidden", but never mind for the puposes of this problem!)

Answer 1