King's Cross - the Heisenstation

I've been working on a little theory. Non-British readers might be interested in this as a piece of cultural information, particularly if they're planning to be in London for the 2012 Olympics. Be warned - there will be math! You don't need to understand it to get my drift, though. Skip any bits that look too tough, it won't matter.

Regular rail travellers between East Anglia and London will have noticed that at King's Cross station, platform assignments for the outbound trains can be a little...random. Not only that, the assignments are often made with only about five minutes to go before the scheduled departure time.

Now, it's easy to put this down to simple incompetence. After all, on the European continent, timetables routinely show both the time and - crucially - platform of each departure. And these aren't just terminals, most are through stations dealing with a massive volume of traffic, and they can be handling trains from neighbouring countries too. Compared to that, dealing with a much smaller number of arrivals at a single-country^*1 terminus ought to be easy, yes?

Well, it probably is. So why can't King's Cross get it right? I believe the answer lies not in hopelessly inept staff and incompetent rail networks but in quantum physics.

In brief: King's Cross is an Heisenstation. Or Heisenbahnhof. Or both.

If you have any familiarity at all with popular quantum physics, you'll have heard of Shrödinger's Cat. If you haven't there's a slightly silly explanation here. It's a(n almost) practical illustration of Werner Heisenberg's^*2 Copenhagen Interpretation of quantum indeterminacy. (Let's leave it as a thought experiment. It might get you in trouble with the animal rights activists if you tried it out for yourself.)

In our Heisenbergian imaginings, King's Cross exists in a superposition of states for any given train planned to depart, each state corresponding to a possible platform. If all platforms were available, the state equation would be:

|ψ> = _p=0∑¹¹ |θ_p>

Keen readers will note that this appears to start from platform 0, not platform 1. Yes, that's right. King's Cross (KGX) acquired a "platform zero" a few years ago, to the delight of mathematicians and programmers everywhere.

However, King's Cross is a working station, and moreover it's English, so you can almost guarantee that some platforms will be unavailable. Let's define the vector of excluded states (platforms) as:

|χ> = _p=0∑¹¹ (|θ_p> if unavailable(p))

So our final superposition is:

|ψ> = (_p=0∑¹¹ |θ_p>) - |χ>

(Bonus assignment: do this using state probabilities instead.)

So much for the math. Now some more concrete stuff. (There's too much concrete around KGX already, but we'll let that aside.)

Normally, humans are considered to be quantum observers, whose observation during the progress of an experiment collapses the probability wave function and settles it to a final value. In other words, the presence of people waiting for the platform indicator to show the assigned platform for a given train ought to settle the platform assignment immediately. This leads us to a startling assertion:

Passengers do not count in, and have no relevance to, the operation of King's Cross

A little contemplation, combined with bitter experience, will quickly confirm this prediction by practical demonstration.

Which leads us to the question: how does that wave function collapse? How do we ever achieve a platform assigment?

It is my contention that the qualified observer must in fact be a person outside the system, someone for whom the platform assignment is a matter of no direct consequence, someone who, in effect, doesn't care.

That would be the station staff.

Extra credit assignment

For extra credit, prove or disprove the statement: at King's Cross, it is possible to know with absolute certainty, at any given instant prior to departure, either a train's platform number, or its actual departure time, but not both.

Next week

In next week's tutorial, we attempt to calculate the overall directed Brownian motion effect on the disabled, slow or elderly of the "platform rush", as more agile and able-bodied commuters jostle past them, competing to secure their seats on the train following the last-minute platform announcement.

We will also explore the likelihood that any given member of those less-able passengers will manage to find a seat once the fast-moving passengers have been seated, taking into account the probability of the late-scheduled train having already departed before they could get to it.

^(*1) The United Kingdom is a country, damn it! I know because every bloody web form in the world says so.
^(*2) Yes, I know, there were others too. Let's not get Borhing about it.