Eigenfunctions of Schrödinger equation Why are solutions of the Schrödinger equation called eigenfunctions? For an electron moving in one dimensional lattice the eigenfunctions are given by$$\psi(x)=u_k(x)e^{ikx}.$
 A: The eigenvalue is something physicists should be familiar with. For some matrix, $A$, multiplied by some vector $\mathbf x$, we get
$$
A\mathbf x=\lambda\mathbf x \tag{1}
$$
where $\lambda$ is the eigenvalue, a characteristic of $A$ on $\mathbf x$.
An eigenfunction is related to Equation (1). Given an operator (a differential operator in the case of quantum mechanics), $\mathcal{A}$, acting on a function, $f(x)$, we have the relation,
$$
\mathcal{A}f=\lambda f\tag{2}
$$
where $\lambda$ is still called the eigenvalue. A function that satisfies this relationship is called the eigenfunction.
Note that not every function satisfies this relationship. For instance, given $\mathcal{A}=\frac{d}{dx}$ (first-order differential operator) and $f(x)=x^2$, the resulting operation is
$$
\mathcal{A}f=\frac{d}{dx}\left(x^2\right)=2x\neq\lambda f(x)
$$
so this does not satisfy (2). However, if $f(x)=e^{kx}$, then
$$
\mathcal{A}f=\frac{d}{dx}\left(e^{kx}\right)=ke^{kx}=kf(x)
$$
which does satisfy (2) with eigenvalue $\lambda=k$. 
