# A photon walks into a hotel…

A photon walks into a hotel and checks in. "Do you want a hand with your luggage?" asks the receptionist. "No thanks", replies the photon, "I’m travelling light".

Thanks to my friend Julie for that one. But it got me thinking about the quantum nature of things that may not immediately appear quantum-like. There’s a neat little rule that says that the classical behaviour of a system can be ‘derived’ from the quantum behaviour by taking the limit of Planck’s constant going to zero. An example is from the Heisenberg uncertainty principle. It tells us that we can’t know position and momentum simultaneously to arbitrary accuracy – the uncertainty in our knowledge of position times the uncertainty on our knowledge of momentum must be greater than Planck’s constant divided by 4 pi. So, if we know the position really well (its uncertainty is low) we can’t know much about the momentum (its uncertainty is high). Now, let’s assume Planck’s constant is given the value zero rather than its real value of 6.626 times 10 to the power minus 34 Joule seconds. That tells us we CAN know position and momentum to arbitrary accuracy (since the product of their uncertainties is bigger than zero), which is the result we are intuitively familiar with in ‘normal’ things. (E.g. the car is travelling through this intersection at 50 km/h – we know both position and speed). It works for other things too – for example, the energy of a photon is given by its frequency times Planck’s constant – if we assume Planck’s constant is zero then photons would carry no energy at all (i.e. be irrelevant) – and light could be described perfectly by a wave. That’s the classical result.

The reason that we don’t experience many quantum effects in everyday life can be put down to Planck’s constant being very very small. To get quantum behaviour showing, you need to look at very small length scales. If you want some theoretical physics fun, have a think about what would happen with everyday things if, say, Planck’s constant were 1 Js. Life would be a bit different – we’d see quantum effects everywhere.

And finally, thanks to one of my third year students…

Heisenberg and Schrodinger are driving in a car and are pulled over by a traffic cop. "Excuse me sir", the policeman says to Heisenberg, "but do you know how fast you were travelling?" "No", replies Heisenberg, "but I can tell you exactly where I was".

The officer is not impressed. "Open the boot", he demands. After a look in, he walks round to Schrodinger. "Did you know there’s a dead cat in the boot?", he asks. "I do now…" replies Schrodinger.

## 2 Responses to “A photon walks into a hotel…”

Hilarious. A question though, assuming a non-zero Planck’s constant – at what sorts of sizes or lengths do these quantum effects become apparent? Are there any examples that we could experience everyday? Or do we need special microscopes or lasers in order to be able to ever actually see them?

Yes, I’ve been a bit ‘loose’ when I said Planck’s constant is ‘small’. Small compared to what?

Put another way: How small do you need to go before quantum effects matter.

To do this construct a quantity with the dimensions of length – e.g., if we are talking about electrons, we’d use the Compton wavelength, h/mc, where h is Planck’s constant (6.6 times 10 to the power minus 34 Joule seconds), m is the mass of the electron (9.1 times 10 to the power minus 31 kg) and c is the speed of light (3.0 times 10 to the power 8 metres per second). This gives a length of 2.4 times 10 to the power minus 12 metres (2.4 pm, or 2.4 billionths of a millimetre.) That’s small. If we were dealing with the more massive proton, not an electrons, the distance would need to be about a thousand times smaller still, or, a rather massive person, (say 60 kg), then we would end up with an exceptionally tiny value.

If we were to take Planck’s constant to zero, all Compton wavelengths would go to zero, and you’d have to get to infinitessimally small lengths before quantum effects mattered.

But, boost it to 1 Js, as I said, then the Compton wavelength of say a 0.1 gram (?) insect would be 1 J s / (10^-4 kg x 3 x 10^8 m/s) = 3 x 10^-5 metres – small but not all that small. Real things (like insects) would have distinct quantum behaviour… Swatting a quantum fly would be frustrating