A couple of years ago I wrote about some things I was thinking about regarding the use of complex numbers in quantum mechanics. This past week I got to refer to that post to my Modern Physics students, and this week we’ll work on the suggestion I made at the end of that post
I think interpreting those two equations is of interest as well. Essentially the spatial curvature of one produces temporal changes in the other and vice versa. That’s actually pretty cool as you could stare at a snapshot of both the two and predict what’s going to happen in the next moment of time.
Last week it was fun to ask my students about how weird they thought wave functions were when they considered that they were complex. We really haven’t gotten to the Schroedinger equation yet, but we have talked about de Broglie waves, and I mentioned that assuming they were cosine or sine waves was problematic from the probability perspective ( as a probability density has a bunch of spots where you couldn’t find the so-called free particle). I showed them how works out better in that regard, and then we had the conversation about the use of complex numbers.
Some of them had seen complex numbers used when talking about waves before. Of course, in those cases, it’s always used as an accounting trick, where we throw away the crazy imaginary parts at the end of the calculation. Here we seem to need to keep them.
I mentioned my old blog post, and we spent some time exploring why complex numbers bother them so much. (Note, at least one student made it clear that it didn’t bother him at all.) Mostly we landed on the fact that there’s so much cultural baggage around the word “imaginary.” So we spent some time exploring other ways to think about it.
I asked whether it would ever make sense that a farmer had 3+2i cows (I asked another first year seminar class a similar question, once, where we were talking about how human consciousness is going to change in the future). We decided that didn’t make any sense. Then I asked if it might make sense if I said I had 3+2i “pencils-and-pens” (spoken fast together). They said that they could interpret that as 3 pencils and 2 pens. Then we got talking about things that often get partnered up. One student suggested hot dogs and buns. What do you think? Does it make sense to say 3+2i “hot dogs and buns?” We were buying it, anyways.
What was interesting about that conversation is that I kept bringing it back to the wavefunction, asking if using the notion of keeping track of two things seemed less weird that a “complex” wavefunction. It seemed that it was.
I did point out the less-than-supportive comments on my last post on this. I was quick to tell them that some of those could be dispensed with if I admitted that the complex number approach is certainly a very handy way of doing things. But right now I care more about them buying into the fact that something is waving than knowing how to calculate things. That’ll come later.
Some of the other comments were also interesting. One talked about other non-spatial eigenfunctions, and, I agree, those are harder to deal with. Another talked about how you can get separate equations for the amplitude and phase of the wavefunction. I think that’s cool, but I’m really excited for the spatial stuff we can do this week using real and imaginary parts (see the quote above).
So, what do you think? Do you agree with our text that “you can’t ask what’s waving?” Do you ever say “wow, our universe has real and imaginary parts!”? What did you think about when you first learned about this?