My mind blew the day I learned about the existence of imaginary numbers. My high school teacher nonchalantly just brought them up as something that was invented to give an answer to what the square root of a negative number was. I imagine most students file that information under ‘irrelevant things I will never use in real life.‘
Complex numbers have many applications in many fields, mostly via Euler’s theorem. But the prevalent formulation of Quantum Mechanics fundamentally relies on the existence of complex numbers. If we take time dependent Schrödinger’s equation, one realizes that a wave function must necessarily be a complex function hiding all the real information of a physical quantum system.
We know the right hand side of the equation has to be real, for we want to obtain real eigenvalues for our observable quantities, therefore we use–or rather construct–observables as Hermitian operators. If Ψ was real, wouldn’t there be a contradiction with the left hand side which already bears an i? So Ψ has to be complex by nature, could that be related to the reason why it doesn’t have a physical interpretation? In the end, just like with any complex number, we leave it to their norm to speak for them by taking the product Ψ*Ψ or |Ψ| and interpret it as a probability density.
So, I find it amazing that a complex entity such as Ψ, with no physical interpretation, yields real observable quantities via the eigenvalue equations. File this post under ‘posts that should have been a Tweet.’