Complex Amplitude: Physical Meaning & Significance

[MadRocketSci2]

I am curious about the complex nature of the amplitudes used in Schrödinger's equation, and what (if anything) it means physically.

I ran across this topic being discussed sometime in the past on this forum, and several posters alluded to some deep significance to this fact, but didn't provide anything specific.

I'm not seeing it so far. It appears to me that the complex valued amplitudes are just a mathematical shortcut and don't appear to have any physical meaning beyond that. It appears to basically reduce the usual second order wave equation with two fields to a 1st order wave equation with one field. You still have all the same degrees of freedom, they're just pasted together in the same variable, taking advantage of linearity. (A differential equation version of the EE trick for handling sin-wave signals).

The dot product of the complex field is the same as the sum of the dot products of the two independent real fields of the second order equation. Nothing new there.

I suppose the real significance would be if you were dealing with a nonlinear equation. In my reading so far, it is claimed that the laws of QM are linear in all cases measured so far. If we were to ever encounter a situation with nonlinear behavior, I would anticipate the nonlinearity to look different operating on a complex variable, versus two real ones. (Either that, or you would just come up with different looking operators for each case to reproduce the same behavior in the model.)

So what's the significance? Where does mathematical convenience end and a concrete statement about the operations of physical law begin? (Or is it all mathematical convenience, using higher DOF mathematical objects at all in the first place?)

 

[Ken G]

I'd say there's no obvious distinction between what is physically significant and what is a mathematical convenience. Conservation of energy, for example, is just a mathematical convenience-- we define energy to be whatever it needs to be to be conserved. But you can hardly find a more profound physical principle! What is profound about quantum amplitudes is that they can interfere, so we replace the classical idea that something that can happen in more than one way is more likely to happen with the possibility that it could be less likely by virtue of having multiple ways of happening. Expressing that profundity by using complex ampitudes is just a mathematical convenience-- and the World Cup championship is just a football game.