Bill Menke's BLOG Page: The Limits of Explanation
As a teacher, I often have to respond to questions that start 'Professor, why is it that ...". These questions are perfectly fair; they indicate to me that a student believes that there is something rather too arbitrary about an assertion I've made in class. But answering them is often rather tricky. Usually, I try to relate the subject with something with which the student is more familiar, an therefore thinks is more normal and less arbitrary.
A particularly interesting question of this type was asked of me recently, by a first-year undergraduate taking Columbia's Frontiers of Science course. I was teaching about quantum mechanical wavefunctions that week, and the question was, "Why do you have to square of the wavefunction to get probability?". Obviously, the student thought computing a square was rather too arbitrary. Why do you have to do that?
A TA would have just said, "That's the way that Erwin Schrodinger said that is was to be done", and left it at that. I thought about it a while, and came up with something different: "That choice is pretty much equivalent to conservation of mass*, which is a pretty fundamental law of the universe. Even though quantum mechanics says the position of a particle is fuzzy - governed by probabilities - there is still exactly one particle there all the time. The mathematics of quantum mechanics indicates that the square of the wavefunction, and not its fourth power or its square root, has this property".
I'm still mulling over whether this is really a better explanation, or whether it just raises more questions. A lot of what we scientists put forward as answers are really just connections. Is conservation of mass really all that obvious? Is the claim that Schrodinger's equaion can link its conservation to a squared quantity really helpful to someone without advanced training in math?
*Well, more precisely, total probability. Here are the details. Do they help?