I've been thinking about how the Schroedinger equation intimidates students because it is inherently complex:
(note the red i). This has led me in the past to say things like "quantum mechanics is weird because apparently the universe is both real and imaginary" and "we can only see the real parts". Now that last quote especially is suspect since what we can "see" is really the magnitude squared of the wavefunction:
which corresponds to the probability density of finding something in a particular location. But I'm coming around to a position where even the first quote above is suspect. It seems to me that you can rewrite the equation as two coupled equations:
Note how the 1's and 2's switch places. What I've done here is renamed the real part of as and the imaginary part of as . Only here I'm just numbering them and not really giving any preference to either. These two equations are totally equivalent to the original equation and what it says about the universe is that for every object you need to keep track of two things. In the end to make predictions about the object you'll need but note that I don't need imaginary numbers at all!
I think interpreting those two equation 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.