# Classical results proved using quantum mechanics

Are there any results in classical mechanics that are easier to show by deriving a corresponding result in quantum mechanics and then taking the limit as $\hbar\rightarrow0$?

(Are there classical results that were first discovered through taking the classical limit of quantum mechanics even if they are now easier to demonstrate, in hindsight, using classical mechanics?)

Update: One semi-example is the derivation of classical adiabatic invariants from the quantum adiabatic theorem in some textbooks. But the quantum adiabatic theorem isn't really any easier.

• Just being pedantic here, but as far as I understand the correspondence between classical and quantum mechanics is a bit more subtle than taking the limit hbar -> 0. See arxiv.org/abs/1201.0150 for some interesting elaboration on the topic. Jun 10, 2013 at 5:13

## 2 Answers

It frequently happens in classical relativity that it's much easier to carry out certain reasoning by talking about photons.

An example of some historical interest is in Einstein's 1905 paper on SR, section 8, "Transformation of the Energy of Light Rays. Theory of the Pressure of Radiation Exerted on Perfect Reflectors," where he says, "It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law." If you already know $E=h\nu$, then you know trivially that $E$ and $\nu$ have to transform the same way. Since Einstein was establishing it for the first time, he had to go through a lengthy argument about the transformation of the electric and magnetic fields. I think this was around the same time he was working on the photoelectric effect, so he must have realized that this was necessary if his theory of light quanta was to be consistent with relativity.

A similar example is the derivation of gravitational time dilation using the standard thought experiment about a photon being emitted and received at the floor and ceiling of an elevator. I find it a lot easier to talk about this example by talking about a photon, although you can certainly get the same result without quantum mechanics.

• Your illustration with Einstein theory is really interesting, since it is exactly the same argument (reversed) that Broglie used to propose $p=\hslash k$: he just wanted a relation relativistically covariant. My comment has nevertheless nothing to do with the question... sorry for that. To come back to the question, I think the Maxwell demonstration of radiation pressure is better understood / easier demonstrated using the photon concept, too. Jun 28, 2013 at 23:25

I just came across an example that I'll post as an answer to my own question.

In Brillouin scattering sound passes through a medium causing periodic variations in refractive index and hence forms a diffraction grating that can scatter light. Because the sound waves are moving, the scattered light is doppler shifted. All of these effects can be described classically and so there is a classical argument to determine the frequency of the scattered light. Nonetheless, most discussions of Brillouin scattering I've seen treat the effect as an interaction between phonons and photons. For example, if an incoming phonon has frequency $\omega_s$, and the incoming and outgoing photons have frequencies $\omega_i$ and $\omega_o$ then conservation of energy gives

$\hbar\omega_o = \hbar\omega_i+\hbar\omega_s$.

We don't even need to take the limit as $\hbar\rightarrow 0$.

The classical argument gives the same result but the quantum argument is much simpler (once you already know how to quantise the relevant systems). Higher order scattering from the grating can be viewed as $n$-phonon interactions.