# Proof of the Principle of Superposition [duplicate]

Is there any Proof of the Principle of Superposition? Is it just a principle or is it verified experimentally?

• Being more specific is of utmost necessity... – ubuntu_noob Jan 5 '17 at 6:50
• I'm not keen on this trend of drive by downvoting that seems to have developed recently. This seems a perfectly good question and doesn't deserve a downvote. It's a beginners question, but a fair one. – John Rennie Jan 5 '17 at 6:59
• It's not an obvious duplicate, but you should have a look at If we can prove that superposition exists, then why can't we measure it? as I think it more or less answers your question. – John Rennie Jan 5 '17 at 7:02
• @JohnRennie: And surprisingly, that is closed under unclear what you're asking – ubuntu_noob Jan 5 '17 at 7:06
• you mean the superposition of quantum mechanics?If so, then it is the fundamental axioms, so there has nothing to do with proof. – Jack Jan 5 '17 at 7:13

There are many different versions of the principle of superposition, depending on the area of physics; the two most common are superposition in electromagnetism and in quantum mechanics.

In all cases superposition comes about when the physical quantity is represented by a function $f$ that satisfies an equation of the form

$$L(f) = g,$$

where $L$ is some operator and $g$ a given function, which may be zero; we typically interpret it as some kind of "source" for $f$. For example, the Gauss equation in electrostatics is of this form, with $f$ the electric field $\mathbf{E}$, $L$ the divergence, and $g = \rho/\epsilon_0$. The crucial property is that $L$ be linear, which means that for any two functions $f_1$, $f_2$ and any real number $c$ we have

$$L(f_1 + c f_2) = L(f_1) + c L(f_2).$$

Suppose $f_1$ satisfies the equation with source $g_1$ (so $L(f_1) = g_1$) and $f_2$ satisfies the equation with source $g_2$. Then $f_1+f_2$ satisfies the equation with source $g_1 + g_2$, since

$$L(f_1+f_2) = L(f_1) + L(f_2) = g_1 + g_2.$$