Hi user- first of all, that isn't quite the situation that I was thinking of. But actually, you're correct- it is impossible to have two real delta functions moving towards each other! A wavefunction must have a varying phase to be moving- take a look at the definition of a probability current: en.wikipedia.org/wiki/Probability_current

However, here is what I had in mind when I made that statement: let's imagine you have a beam splitter, like one of the ones in the diagram in my answer, that has two possible input ports and two possible output ports. If you shine light into input A or B, you get 50% out one port and 50% out the other.

Now, you might think at first that you can use this beamsplitter to violate energy conservation. Here's why: if you shine a beam of light into one port with energy E, you get out energies 1/2 E from each output. The energy in a beam of light goes as the strength of the electric and magnetic fields squared, or $E=|E_0|^2+|B_0|^2$ (ignoring some constants). So a natural idea is just to shine light in from both the ports.

Light also obeys a superposition principle, so the amplitudes at any point just add. This means that shining light in both ports gives you twice the amplitude of light anywhere, and thus 4x the energy, that you have from one beam. And you only sent in 2E! Seems too good to be true, right?

Of course, it won't really work out that way. What really happens is that you matter how you try to split and recombine the beams, how you design your beamsplitter or what fiber optics you try to use or whatever to get the light aligned, exactly half of it will destructively interfere and half will constructively interfere, so that you get out exactly the energy you put in.

This also tends to seem like a "conspiracy" when you first encounter it, but it is how things have to work for physics to be consistent. And I've intentionally phrased all this in classical terms, but the analogy with QM is exact- just turn down your light sources until they emit one photon at a time and write the modes of the beamsplitter in terms of states, and you can identify the field strengths are being (proportional to) quantum amplitudes and energies as (proportional to) probailities.

That is a very good example, and you're right it is a good analogy. But it still leaves me with a dilemma, which is that I'm exasperated that no one seems to empathize with what I'm looking for. A big criticism of the MWI is that the measure (ie the "squared" in the Born rule) is unintuitive and contradicts naive branch counting.

So my goal was to just be a good student and understand this criticism by constructing a simple example that shows something where for example you have branches in a 1:2 ratio but that the Born rule tells you the probabilities are 1:4.

It seems that I've been foiled in that attempt, and I'm not sure if that is because the criticisms of the MWI in this regard are actually wrong (as proponents of the MWI claim) or just that I am not doing a good enough job of asking the question.

In any case I would be really grateful if you had any suggestions regarding my dilemma or how to re-pose the question in a way that might get answered!

Thanks for your help!