### Ignoring non-sensical solutions

Warning: Contains simple puzzle spoilers!

On and off I've been working through Rae's Quantum Mechanics. One of the things that annoys me is the way the solutions to problems are chosen. Take a particle in a potential well: A differential equation is given, solved, physically non-sensical solutions are discarded, and what remains must be the solution.

How on earth can we justify only considering the 'sensible' solution? Can we really claim we have a mathematical model if it involves such judgement calls?

It reminds me of a simple mathematical puzzle. We have a cell, which, every time step, dies with probability 1/3, or otherwise splits into two cells, each of which follow the same rule in future timesteps. The children behave independently. What is the probability it survives forever?

If x is the probability it doesn't survive forever, this is because it either dies in the next step, or both children must die - as they are independent, the probability of this is x^2. So, x = 1/3 + 2/3 x^2. Solving this gives x = 1 or 0.5. Ignoring x = 1, we have the answer x = 0.5. But... why can we ignore the incorrect solution of certain death?

One way to look at it is as a convergence issue. If you plug in numbers around 0.5 into 1/3 + 2/3 x^2, you get a number closer to 0.5. If you plug in a number just under 1, it'll end up further away from 1. This shows that 0.5 is a 'stable' answer, but 1 isn't.

However, this still isn't convincing. What you can do is define x_i as the probability of the colony dying out in i steps, with x_{i+1} = 1/3 + 2/3 x_i^2. We can then say the colony death probability is the limit as i goes to infinity, and show that the limit is 0.5, if x_0 is in the range [0,1), thanks to the above convergence properties, and solving the quadratic is simply a way of finding the converged value for the real solution.

In a similar fashion, I believe the differential equation isn't the real quantum problem. It's an equation that represents possible equilibrium solutions, but misses out how to select the real one from all the possibilities it brings. Somehow, there is a mathematical phrasing for the real problem we're trying to solve, one that eliminates the 'non-sensical' solutions automatically, but I'm not sure what it is.

Posted 2012-03-24.