Mathematics and Science Troll Questions

This is meant to be a list of questions with particularly subtle answers. You may or may not find them difficult.

In chemistry, we often talk of negative and positive parts of the wavefunction. But the wavefunction is complex-valued, how could this possibly be legal?

For an even function f(x), why is the following incorrect? (Integrating over all real numbers)

(Note that d(-x) = -dx!)

Neutrinos are always created as a "mixture" of the "real" ν1, ν2 and ν3 particles (the "mass eigenstates"). Suppose that a neutrino beam is created somewhere and detected somewhere else. The detector will detect a phase shift between the different species. To calculate that phase shift, note that the phase shift of a neutrino travelling from the source to the detector is simply given by m τ where m is the mass and τ is the proper time (working in natural units). Since proper time obeys τ = t/γ and γ = E/m the phase shift should be given by m2 t/E. However the correct answer is 1/2 of that. Where did we go wrong?