Why is the Ritz combination principle incompatible with Classical Mechanics? This is a quote from Dirac's Principles of Quantum Mechanics:

"(...) if an atomic system has its equilibrium disturbed in any way
  and is then left alone, it will be set in oscillation and the
  oscillations will get impressed on the surrounding electromagnetic
  field, so that their frequencies may be observed with a spectroscope.
  Now whatever the laws of force governing the equilibrium, one would
  expect to be able to include the various frequencies in a scheme
  comprising certain fundamental frequencies and their harmonics. This
  is not observed to be the case. Instead, there is observed a new and
  unexpected connexion between the frequencies, called Ritz's
  Combination Law of Spectroscopy, according to which  all the
  frequencies can be expressed as differences between certain terms, the
  number of terms being much less than the number of frequencies. This
  law is quite unitelligible from the classical standpoint."

I'm having trouble understanding this paragraph. Assuming that the atom is a system in equilibrium that emits e-m waves when perturbed and these e-m waves are product of the oscillations of the atom about its equilibrium configuration that result from the perturbation, does it follow that the Ritz's law is in contradiction with classical mechanics? Why? Thanks.
 A: In classical mechanics, you can make up a complicated system with many different natural frequencies. In general, these frequencies are completely independent of each other. Due to non-linearities in the coupling forces, it may happen that when two modes are vibrating simultaneously, you get a new frequency appearing in the spectrum as the sum or difference or the two primary modes. But in quantum mechanics, you never see the primary modes at all...you only see the sum or difference frequencies. Furthermore, if you try to explain them by non-linear forces, you should also expect to see multiples of the fundamental frequencies. These are absent in, for example, the spectra of atoms. It's hard to explain by a classical model involving things like masses and springs. It manifests itself in QM, of course, because the "fundamental" frequencies, the natural modes, evolve in time without any oscillating charges associated with them. The oscillating charges only appear when you have the superposition of two fundamental modes. This is how quantum mechanics is very different from classical.
A: An example of the Ritz combination principle would be if you have an atom with three energy states, labeled 1, 2, and 3. There are three emission lines, corresponding to $2\rightarrow1$, $3\rightarrow2$, and $3\rightarrow1$. Their frequencies are related by $f_{3\rightarrow1}=f_{2\rightarrow1}+f_{3\rightarrow2}$. This relationship is impossible to understand classically, but quantum-mechanically it simply happens because of energy conservation along with $E=hf$.
A: The fact is that a simple classical explanation of the emission of spectral lines does exist provided that we discard the concept of a photon being an indivisible entity and assign all quantisation to the atom.  In a 'pure state' the atom has no dipole moment and so does not radiate.  If it is in state $N$ and this state is perturbed by a suitable small field it will start to radiate having gained some dipole moment.  It will radiate EM field and decay until it reaches another 'pure state' $M$.  The difference in energy can be expressed in terms of the difference in frequencies ($\Omega_N$ and $\Omega_M$) of the rotating fields of the electrons in their orbits and comes to $h(\Omega_N - \Omega_M)$.  All this has been convincingly explained by Ed Jaynes in a number of papers but see his "Survey of the Present status of Neoclassical Radiation Theory"  Pages 60-61. You will get it on a Google search for the title. 
A: I'd like to weigh in on this very old question by making explicit the simple mathematical formula behind one portion of Dirac's quote: "... all the frequencies can be expressed as differences between certain terms, the number of terms being much less than the number of frequencies. This law is quite unintelligible from the classical standpoint." The answer by @user4552 comes closest to this point but misses the "much less" target, by choosing an example with only 3 energy states.
Suppose we have an atomic system with $N$ energy modes. Each of those modes has a certain energy level. Fixing some base energy level to be $0$, one can write these energy levels as $N$ independent numerical values $$e_1,e_2,e_3,...,e_N
$$
For $1 \le i < j \le N$ the frequency of the photon emitted when jumping down from energy level $e_j$ to energy level $e_i$ is given by solving for $\nu$ in the equation $e_j - e_i = h \nu$; let me denote that frequency as $\nu_{j \to i}$. The number of different frequencies $\nu_{j \to i}$ observed is therefore equal to the number of ordered pairs of integers $i,j$ subject to the constraint $1 \le i < j \le N$, a number equal to
$$(N-1)(N-2)=N^2-3N+2
$$
This number of frequencies grows quadratically as a function of $N$, and is therefore much larger than $N$. Turning this around, this "number of terms" $N$ is, as Dirac says, "much less than the number of frequencies".
The "unintelligibility" of this is that we are observing $N^2 - 3N + 2$ frequencies which ought to be independent numbers, but in fact those numbers seem to depend only on a much small number $N$ of hidden quantities.
