COMPLEX EXPONENTIALS

In your first exposure to SIMPLE HARMONIC MOTION and WAVES you probably saw only the real sinusoidal functions $\sin \theta$ and $\cos \theta$ (where $\theta = kx - \omega t + \phi$, the phase of an oscillation). This was reasonable enough, since all the phenomena of classical mechanics are in fact real, at least in the mathematical sense. Whether they are real in the colloquial sense is subject to discussion....

In QUANTUM MECHANICS, which we claim describes the way the real world really works, things are not always real in the mathematical sense. Well, "things" are always real, if by "things" you mean physical observables, but the things you have to talk about to make predictions about the real "things" -- or at least about what you are likely to measure if you observe one -- those things are not real; they are almost always complex. Sort of like that sentence, eh? No, mathematically complex. That is, complex in the mathematical sense, i.e. having a real part and an imaginary part.

With that introduction to QUANTUM MECHANICS I should have produced the proper state of confusion one needs to approach the subject. But for now I would like to demonstrate a few simple properties of the most remarkable function ever invented: the exponential function, $\exp(x) \equiv e^x$.