# 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.

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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.

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+1 Thanks. It was a very clarifying answer. – becko Jul 28 '11 at 1:39
I'm glad you liked it. Of course, it's only my interpretation of what Dirac might have meant; I don't know what he was really thinking. – Marty Green Jul 28 '11 at 5:56

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 $E=hf$.

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I understand how the Ritz combination principle comes about naturally in Quantum Mechanics... You say: "This relationship is impossible to understand classically". This is what I don't understand. What's the justification for claiming that the Ritz combination principle cannot fit with classical mechanics? – becko Jul 27 '11 at 19:36
People knew the size of the atom and that there are protons in the nucleus (Rutherford). In the classical picture the electron would have to emit electromagnetic waves because it is an accelerated charge. In the classic theory this dissipation entails that the electron would crash into the nucleus after a very short time. A stable "state" can't be explained without quantum mechanics. – whoplisp Jul 27 '11 at 21:16
whopslip, while what you say is true, it does not address the question, which is "Are systems in accordance with Ritz's law incompatible with classical mechanics". Similarly, Ben's answer merely repeats the question as a statement. – BebopButUnsteady Jul 27 '11 at 21:25
I suppose it's difficult to prove a negative, in the sense that what's being asserted is that no sensible interpretation exists for a certain phenomenon within a certain theory. Nobody had been able to dream up a classical interpretation for the Ritz principle by the time Dirac was writing, and AFAIK nobody has since then. It seems extremely implausible that anyone could, since classically there isn't even a reason for the spectrum to be discrete. But of course it's always conceivable that it could happen. – Ben Crowell Jul 27 '11 at 21:32

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.

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