counting normal modes

Question

I'm not sure if my confusion is substantive or merely semantic. So here's the most naïve way to frame it: A free $N$-atom molecule has $3N-6$ vibrational normal modes, with each mode having fixed energy for a given molecule. Classically, the state of the molecule (in the harmonic approximation) can be expanded on the basis of its vibrational modes. In quantum mechanics, the Hamiltonian can similarly be reduced to a sum of $3N-6$ harmonic oscillators. But each of these oscillators has its own spectrum of fixed-energy stationary states. These too are normal modes, since they do not evolve into one another. But they're clearly not the $3N-6$ referred to classically (although they do set the scales for the $3N-6$ quantum spectra). So how does one understand how a finite number of fixed-energy classical normal modes becomes an infinity of qm modes of different energies?

Answer 1

If you consider the classical system then the analysis gives us a set of normal modes, but it does not specify the amplitude of each mode. That is, all the $3N-6$ modes have fixed frequencies but can have arbitrary amounts of energy.

In the quantised oscillator the situation is the same except that now the energy of each mode is quantised. There are still $3N-6$ modes, each with some fixed frequency, but the energy of each mode can only be a multiple of $h\nu$.

So you can't say the quantised system has an infinite number of modes, just that each mode can (in principle, obviously not in practice) have an infinite number of different energies. The classical system also has an infinite number of different possible energies associated with each mode. In fact if the energy is a real number not a rational the classical oscillator has a transfinite number of different possible energies for each mode.