> Figure 4: The wrong wave function! Even though it appears as though this wave function shows two particles, one trailing the other, similar to Fig. 3, it instead shows a single particle with definite speed but a superposition of two different locations (i.e. here OR there.)
I understand that if treat the act of adding two particles' wave functions as creating a new wave function for one particle, then we have this problem, essentially by definition. But it got me thinking - would it not make sense to treat the result as an expected value, such that we could then measure how many particles are likely to be to the right of the door at each point in time?
It isn't by definition, presuming the relationship between quantum mechanics and reality. You can have a _two particle_ state and a _one particle state_ with non-trivial probability of being in two places. They are distinct things. The key idea here (and really, in Quantum Mechanics generally) is that superpositions are important things in the theory. This is the statement that if you have a wave function for one situation and another wave function for another than the sum of the two is also, necessarily, a valid wave function for a physically realizable system.
This is different from a classical probability. Suppose we simply don't know whether the baseball was fired from HERE or from THERE. In a classical situation, we can carry forward our understanding of the situation in time by simply calculating what the classical particles would do independently. In quantum mechanics the mechanics are of the wave function itself, not of the things we measure. We cannot get the right answer by imagining first that we measure the particle in one location and calculate forward and then by imagining we measure the particle in another and calculating forward and then adding the results. It isn't how the theory works. We must time evolve the wave function to predict the statistical behavior of measurement in the future.
> Figure 4: The wrong wave function! Even though it appears as though this wave function shows two particles, one trailing the other, similar to Fig. 3, it instead shows a single particle with definite speed but a superposition of two different locations (i.e. here OR there.)
I understand that if treat the act of adding two particles' wave functions as creating a new wave function for one particle, then we have this problem, essentially by definition. But it got me thinking - would it not make sense to treat the result as an expected value, such that we could then measure how many particles are likely to be to the right of the door at each point in time?