Vibrational spectroscopy energy spectrum I have a question regarding vibrational spectroscopy. In vibrational spectroscopy we are describing the vibration of molecules with the Morse potential which gives us stationary wavefunctions that give us the probability amplitude for finding the whole atom at a certain distance away from the other atom. The energys corresponding to this wavefunctions are discrete and so the spectrum is also discrete. However if the molecule is classical vibrating it should be described by a superposition of eigenstates and hence the spectrum should be continuous. So where is the problem?
 A: Why should the molecule be described by classical mechanics? Classical mechanics is an approximation of quantum mechanics that works well for larger objects, but not at the scale of molecules. And in quantum mechanics the energy values a bound system can have are discrete (see https://en.wikipedia.org/wiki/Energy_level)
A: 
However if the molecule is classical vibrating it should be described by a superposition of eigenstates and hence the spectrum should be continuous. So where is the problem?

Superposition of eigenstates is fine, but it does not imply absorption spectrum should be continuous. When you analyze interaction of external field with the Hamiltonian, it is always strong for resonant frequencies defined by difference between Hamiltonian eigenvalues, and much weaker for frequencies that are not close to any such resonant frequency. Superposition state only changes which resonant frequencies can be "seen" in that state, but it does not introduce continuous spectrum. For continuous spectrum, one needs to have continuous spectrum of the Hamiltonian.
