Is superposition a valid assumption in classical systems?

[elduderino]

I was reading Merzbacher and it is written that it is a fundamental mathematical assumption. I was doubting if it is an assumption at all.

We get the principle of superposition when we solve certain kinds of differential equations (second order, first degree or linear) where we get as solutions linearly independent functions which span the functional basis $$\{ \sin x,\cos x \}$$ or $$\{ e^{ix}, e^{-ix} \}$$. It is by nature of these solutions that the entire basis can be spanned, and by the linear nature of the differential equations that any linear combination of this basis set can be represented as a solution.

Why does Merzbacher still call it a mathematical assumption? Where am I wrong in my understanding.

 

[dextercioby]

My view on this topic is that <linearity> is a fundamental assumption of a mathematical and physical nature. Denying it means rejecting all other postulates and replacing everything with new mathematical equations. Schroedinger's equation would have to be replaced.

A consequence of the Hamiltonian operator being linear on a Hilbert space is that the linear combination of the solutions to S.E. is also a solution. Linearity is postulated and stems from every assumption.

 

[paultant]

In quantum mechanics, the principle of superposition is of central importance. Dirac says, it requires us to assume that between states of a system there exists peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The origin state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any states may be considered as the result of a infinite number of two or more states, and indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state.

 

[Pythagorean]

I was reading Merzbacher and it is written that it is a fundamental mathematical assumption. I was doubting if it is an assumption at all.

We get the principle of superposition when we solve certain kinds of differential equations (second order, first degree or linear) where we get as solutions linearly independent functions which span the functional basis $$\{ \sin x,\cos x \}$$ or $$\{ e^{ix}, e^{-ix} \}$$. It is by nature of these solutions that the entire basis can be spanned, and by the linear nature of the differential equations that any linear combination of this basis set can be represented as a solution.

Why does Merzbacher still call it a mathematical assumption? Where am I wrong in my understanding.

In classical systems in nature, superposition does not hold in general. Let's say you have two inputs (x1 and x2) to a system (the weather or a brain, for instance). The system does some analysis or transformation (T) on the input and gives an output for each (y1 and y2).

When superposition holds:

T(x1 + x2) = T(x1) + T(x2) = y1 + y2

but in general, you can't assume that, because of the way information propagates through physical systems (at the classical level). x1 and x2 interact with each other. They can amplify or dampen each other, so they don't exist as simple superpositions, easy to separate: they are couped together and thus:

T(x1 + x2) != T(x1) + T(x2)

instead:

T(x1 + x2) = y3

(y3 is a solution that usually has to be found numerically, since analytical solutions seldom exist for such systems).