The soul of quantum mechanics, to me, is the complex number system.
When we look around us, we see what mathematicians call “real three-dimensional space”, at least that’s what it looks like. Who knows what it actually is, but that’s what it looks like.
In much the same manner, quantum mechanics looks like complex space, of some kind. The more I study quantum mechanics, the more it looks to me like complex numbers, not real numbers, are the basic kind of number system that it’s made of.
Schroedinger’s Equation, a standard facet of quantum mechanics, gives us a “wavefunction”. A wavefunction is almost like a probability density function (PDF), except that it’s complex-valued, and that makes it like two probability density functions encoded into one.
It’s easy to way to convert a wavefunction to a real function and that’s to take the magnitude of its complex value. That gives us the probability density function of the particle’s position.
It seems, however, that we’ve thrown away some kind of information, specifically the phase information in the wavefunction, and we have. We can extract this information by taking the Fourier transform of the wavefunction, which gives us another complex wavefunction, then we take again the magnitude of this new wavefunction to get the probability density function of the momentum.
Thus, the wavefunction encodes both the probability density of the position and the momentum into one function, and it uses complex analysis to do it!
Based on our knowledge of classical mechanics, this is no real surprise. Newton’s second law tells us that we need to know both the position and the momentum to predict the particle’s motion.
Schrodinger’s equation is pretty simple in its complex fomulation. We could split the wavefunction apart into real and imaginary components and try to express Schrodinger’s equation using them, which is a mess. Perhaps amplitude and phase components might work better, but I haven’t tried.
The simplest and most elegant way to express quantum mechanics seems to be with complex numbers.