Why Are Wave Functions Additive in Quantum Mechanics?

[freddyfish]

The empirical method of linear combination of atomic orbitals is not hard to understand. There is one thing about this method that doesn't make too much sense, though.

Why are wave functions additive? This property of the wave functions is crucial for this method. Claiming that the wave functions have this property is evidently justified by experimental examinations of molecules, but:

-Is there some pleasing answer to why two functions, which nobody really know what they represent unless if their absolute value is squared, are additive?

Thanks //F

 

[jtbell]

Why are wave functions additive?

Schrödinger's Equation is a linear differential equation. Any linear combination of solutions to a linear differential equation, is also a solution of that differential equation.

 

[Matterwave]

Alternatively, one can view the principle of superposition as a postulate on which QM is built. In such a case, the Schrödinger equation is required to be linear.

 

[Jazzdude]

The short answer is that we observe interference and this interference does not create interactions. That means two waves can penetrate each other and come out unchanged. This implies linearity.
Of course we can only measure this up to a certain precision, and so nonlinear quantum theory is not completely ruled out. In fact, there are attempts to solve the quantum measurement problem by postulating a very tiny bit of nonlinear behavior, just enough to mess with macroscopic objects. There are a few non-obvious problems with this approach however.

 

[Ken G]

In addition to the good points that have been made, I would say the reason we need the concept of superposition in quantum mechanics is that we have the bizarre empirical discovery that we can prepare two particles in seemingly the exact same way, then do an observation on them, and get that half come out at X and half come out at Y, and none come out at (X+Y)/2. We can't say we prepared half at X and half at Y, because we can't find any differences in our preparation of the systems that we could use to predict those different outcomes, and we can even get interference between those possibilities (as in a two-slit experiment), so we need some other way to talk about the state of these particles that involves some kind of combination of X and Y (and that's where the linearity comes it, it really doesn't work very easily otherwise). Some believe there are differences and we just can't find them, others feel there are no differences and the superposition is fundamental, but either way it is our inability to distinguish the preparations of these different outcomes that force on us some kind of concept of superposition, and the possibility of interference that forces the particlar type of superposition we find in quantum mechanics. The linearity is important and that's why the wave functions are additive, but the fundamental strangeness of quantum mechanics that underlies superpositions of states is that identical preparations lead to different outcomes that affect each other.