Can light have an irrational wavelength?

Question

The title is pretty straightforward. I was wondering if you can have light whose wavelength is not a rational number, but an irrational one. It seems to me there is nothing preventing this from happening, but I am not sure since I've never been exposed to such an instance! Is it possible for light to have a wavelength of, say 100*pi nm?

Answer 1

I agree with you, there is nothing preventing this from happening, not to mention that if it is rational for a certain unit, it could very well be irrational fro another unit (example new unit = $\pi$ meters). And as the choice of unit is arbitrary...

Addendum: for unit dependent quantities, one can chose units that make a given measure rational or not. But there are other quantities where we have no choice as for example $\pi$ or the proton-to-electron mass ratio (those are dimensionless constants).

Answer 2

Not really, because irrational numbers are a mathematical concept without an exact analogue in the physical world.

The wavelength of light is related to the momentum carried by each photon according to the de Broglie relation $p = h/\lambda$. And for all quantum-mechanical objects, the uncertainties in momentum $\sigma_p$ and in position $\sigma_x$ are related by the Heisenberg principle, $$ \sigma_p \sigma_x \geq \hbar/2.$$

A mathematical way to think about the Heisenberg Uncertainty Principle is to recognize that a pure, single-frequency sinusoidal wave has infinite spatial extent. If you want to confine your wave to some finite region of space, like "our galaxy" or "planet Earth" or "this laser cavity," your wave has not a single frequency but a distribution of frequencies which interfere constructively where your wave "is" and destructively where your wave "isn't."

Therefore if you have any information about the spatial extent of your wave, $\sigma_x < \infty$, you must have some corresponding uncertainty in its momentum and wavelength. A measurement of the wavelength will give you an estimate of the central value and an estimate of the uncertainty on that central value. In an ordinary measurement, you report the central value truncated to some rational number, where the method of truncation is based on the uncertainty. However the set of non-truncated numbers which are consistent with that central value, which you could have reported instead of that truncated central value without changing your meaning, is an infinite set which contains infinitely many rational and irrational numbers.

The importance of harmonics in describing wave dynamics means that there's a stronger case for describing waves whose wavelengths occur in rational ratios, but the same basic problem applies there. Rational and irrational ratios aren't really consistent with a world where precision is finite.

When my laboratory students measure that one thing is one-third the size of another thing, and record "$\text{ratio} = 0.\overline{333}$" in their lab notebooks, that's when we have this little discussion. There aren't repeating decimals in laboratory experiments, and there aren't nonterminating, nonrepeating decimals either.